Hamiltonian (quantum mechanics)

From Wikipedia, the free encyclopedia

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by ${\displaystyle \hat{H}}$, where the hat indicates that it is an operator. It can also be written as ${\displaystyle H}$ or ${\displaystyle \check{H}}$.

Introduction

Main article: Operator (physics) § Operators in quantum mechanics

The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction between particles, kind of potential energy, time varying potential or time independent one.

Schrödinger Hamiltonian

One particle

By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system in the form

$${\displaystyle \hat{H}=\hat{T}+\hat{V},}$$

where

$${\displaystyle \hat{V}=V=V(\mathbf{r},t),}$$

is the potential energy operator and

$${\displaystyle \hat{T}=\frac{\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}}{2m}=\frac{{\hat{p}}^2}{2m}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2,}$$

is the kinetic energy operator in which ${\displaystyle m}$ is the mass of the particle, the dot denotes the dot product of vectors, and

$${\displaystyle \hat{p}=-i\hslash \mathrm{\nabla },}$$

is the momentum operator where a ${\displaystyle \mathrm{\nabla }}$ is the del operator. The dot product of ${\displaystyle \mathrm{\nabla }}$ with itself is the Laplacian ${\displaystyle {\mathrm{\nabla }}^2}$. In three dimensions using Cartesian coordinates the Laplace operator is

$${\displaystyle {\mathrm{\nabla }}^2=\frac{{\mathrm{\partial }}^2}{{\mathrm{\partial }x}^2}+\frac{{\mathrm{\partial }}^2}{{\mathrm{\partial }y}^2}+\frac{{\mathrm{\partial }}^2}{{\mathrm{\partial }z}^2}}$$

Although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes. Combining these yields the form used in the Schrödinger equation:

$${\displaystyle \begin{array}{rl}\hat{H}& =\hat{T}+\hat{V}\\ & =\frac{\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}}{2m}+V(\mathbf{r},t)\\ & =-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(\mathbf{r},t)\end{array}}$$

which allows to apply the Hamiltonian to systems described by a wave function ${\displaystyle \mathrm{\Psi }(\mathbf{r},t)}$. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.

One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.

Many particles

The formalism can be extended to ${\displaystyle N}$ particles:

$${\displaystyle \hat{H}=\sum _{n=1}^N{\hat{T}}_n+\hat{V}}$$

where

$${\displaystyle \hat{V}=V({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N,t),}$$

is the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) and

$${\displaystyle {\hat{T}}_n=\frac{{\hat{\mathbf{p}}}_n\cdot {\hat{\mathbf{p}}}_n}{2m_n}=-\frac{{\hslash }^2}{2m_n}{\mathrm{\nabla }}_n^2}$$

is the kinetic energy operator of particle ${\displaystyle n}$, ${\displaystyle {\mathrm{\nabla }}_n}$ is the gradient for particle ${\displaystyle n}$, and ${\displaystyle {\mathrm{\nabla }}_n^2}$ is the Laplacian for particle n:

$${\displaystyle {\mathrm{\nabla }}_n^2=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_n^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}_n^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}_n^2},}$$

Combining these yields the Schrödinger Hamiltonian for the ${\displaystyle N}$-particle case:

$${\displaystyle \begin{array}{rl}\hat{H}& =\sum _{n=1}^N{\hat{T}}_n+\hat{V}\\ & =\sum _{n=1}^N\frac{{\hat{\mathbf{p}}}_n\cdot {\hat{\mathbf{p}}}_n}{2m_n}+V({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N,t)\\ & =-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}{\mathrm{\nabla }}_n^2+V({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N,t)\end{array}}$$

However, complications can arise in the many-body problem. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:

$${\displaystyle -\frac{{\hslash }^2}{2M}{\mathrm{\nabla }}_i\cdot {\mathrm{\nabla }}_j}$$

where ${\displaystyle M}$ denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as mass polarization terms, and appear in the Hamiltonian of many electron atoms (see below).

For ${\displaystyle N}$ interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function ${\displaystyle V}$ is not simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.

For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle, that is

$${\displaystyle V=\sum _{i=1}^{N}V({\mathbf{r}}_i,t)=V({\mathbf{r}}_1,t)+V({\mathbf{r}}_2,t)+\cdots +V({\mathbf{r}}_N,t)}$$

The general form of the Hamiltonian in this case is:

$${\displaystyle \begin{array}{rl}\hat{H}& =-\frac{{\hslash }^2}{2}\sum _{i=1}^N\frac{1}{m_i}{\mathrm{\nabla }}_i^2+\sum _{i=1}^N{V}_i\\ & =\sum _{i=1}^N\left(-\frac{{\hslash }^2}{2m_i}{\mathrm{\nabla }}_i^2+V_i\right)\\ & =\sum _{i=1}^N{\hat{H}}_i\end{array}}$$

where the sum is taken over all particles and their corresponding potentials; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation—in practice the particles are almost always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by Coulomb interaction (electrostatic force), as shown below.

Schrödinger equation

Main article: Schrödinger equation

The Hamiltonian generates the time evolution of quantum states. If ${\displaystyle |\psi (t)⟩}$ is the state of the system at time ${\displaystyle t}$, then

$${\displaystyle H|\psi (t)⟩=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\psi (t)⟩.}$$

This equation is the Schrödinger equation. It takes the same form as the Hamilton–Jacobi equation, which is one of the reasons ${\displaystyle H}$ is also called the Hamiltonian. Given the state at some initial time (${\displaystyle t=0}$), we can solve it to obtain the state at any subsequent time. In particular, if ${\displaystyle H}$ is independent of time, then

$${\displaystyle |\psi (t)⟩=e^{-iHt/\hslash }|\psi (0)⟩.}$$

The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in ${\displaystyle H}$. One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, a functional calculus is required. In the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient.

By the *-homomorphism property of the functional calculus, the operator

$${\displaystyle U=e^{-iHt/\hslash }}$$

is a unitary operator. It is the time evolution operator or propagator of a closed quantum system. If the Hamiltonian is time-independent, ${\displaystyle \{U(t)\}}$ form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance.

Dirac formalism

However, in the more general formalism of Dirac, the Hamiltonian is typically implemented as an operator on a Hilbert space in the following way:

The eigenkets (eigenvectors) of ${\displaystyle H}$, denoted ${\displaystyle |a⟩}$, provide an orthonormal basis for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted ${\displaystyle \{E_a\}}$, solving the equation:

$${\displaystyle H|a⟩=E_a|a⟩.}$$

Since ${\displaystyle H}$ is a Hermitian operator, the energy is always a real number.

From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the spectrum of an operator). However, all routine quantum mechanical calculations can be done using the physical formulation.

Expressions for the Hamiltonian

Following are expressions for the Hamiltonian in a number of situations. Typical ways to classify the expressions are the number of particles, number of dimensions, and the nature of the potential energy function—importantly space and time dependence. Masses are denoted by ${\displaystyle m}$, and charges by ${\displaystyle q}$.

General forms for one particle

Free particle

The particle is not bound by any potential energy, so the potential is zero and this Hamiltonian is the simplest. For one dimension:

$${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}}$$

and in higher dimensions:

$${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2}$$

Constant-potential well

For a particle in a region of constant potential ${\displaystyle V=V_0}$ (no dependence on space or time), in one dimension, the Hamiltonian is:

$${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+V_0}$$

in three dimensions

$${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V_0}$$

This applies to the elementary "particle in a box" problem, and step potentials.

Simple harmonic oscillator

For a simple harmonic oscillator in one dimension, the potential varies with position (but not time), according to:

$${\displaystyle V=\frac{k}{2}{x}^2=\frac{m{\omega }^2}{2}{x}^2}$$

where the angular frequency ${\displaystyle \omega }$, effective spring constant ${\displaystyle k}$, and mass ${\displaystyle m}$ of the oscillator satisfy:

$${\displaystyle {\omega }^2=\frac{k}{m}}$$

so the Hamiltonian is:

$${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{m{\omega }^2}{2}{x}^2}$$

For three dimensions, this becomes

$${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+\frac{m{\omega }^2}{2}{r}^2}$$

where the three-dimensional position vector ${\displaystyle \mathbf{r}}$ using Cartesian coordinates is ${\displaystyle (x,y,z)}$, its magnitude is

$${\displaystyle r^2=\mathbf{r}\cdot \mathbf{r}=|\mathbf{r}{|}^2=x^2+y^2+z^2}$$

Writing the Hamiltonian out in full shows it is simply the sum of the one-dimensional Hamiltonians in each direction:

$${\displaystyle \begin{array}{rl}\hat{H}& =-\frac{{\hslash }^2}{2m}\left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}\right)+\frac{m{\omega }^2}{2}\left(x^2+y^2+z^2\right)\\ & =\left(-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{m{\omega }^2}{2}{x}^2\right)+\left(-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}+\frac{m{\omega }^2}{2}{y}^2\right)+\left(-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}+\frac{m{\omega }^2}{2}{z}^2\right)\end{array}}$$

Rigid rotor

For a rigid rotor—i.e., system of particles which can rotate freely about any axes, not bound in any potential (such as free molecules with negligible vibrational degrees of freedom, say due to double or triple chemical bonds), the Hamiltonian is:

$${\displaystyle \hat{H}=-\frac{{\hslash }^2}{2I_{xx}}{\hat{J}}_x^2-\frac{{\hslash }^2}{2I_{yy}}{\hat{J}}_y^2-\frac{{\hslash }^2}{2I_{zz}}{\hat{J}}_z^2}$$

where ${\displaystyle I_{xx}}$, ${\displaystyle I_{yy}}$, and ${\displaystyle I_{zz}}$ are the moment of inertia components (technically the diagonal elements of the moment of inertia tensor), and ${\displaystyle {\hat{J}}_x}$, ${\displaystyle {\hat{J}}_y}$, and ${\displaystyle {\hat{J}}_z}$ are the total angular momentum operators (components), about the ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ axes respectively.

Electrostatic (Coulomb) potential

The Coulomb potential energy for two point charges ${\displaystyle q_1}$ and ${\displaystyle q_2}$ (i.e., those that have no spatial extent independently), in three dimensions, is (in SI units—rather than Gaussian units which are frequently used in electromagnetism):

$${\displaystyle V=\frac{q_1{q}_2}{4\pi {\epsilon }_0|\mathbf{r}|}}$$

However, this is only the potential for one point charge due to another. If there are many charged particles, each charge has a potential energy due to every other point charge (except itself). For ${\displaystyle N}$ charges, the potential energy of charge ${\displaystyle q_j}$ due to all other charges is (see also Electrostatic potential energy stored in a configuration of discrete point charges):

$${\displaystyle V_j=\frac{1}{2}\sum _{i\ne j}{q}_i\varphi ({\mathbf{r}}_i)=\frac{1}{8\pi {\epsilon }_0}\sum _{i\ne j}\frac{q_i{q}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}}$$

where ${\displaystyle \varphi ({\mathbf{r}}_i)}$ is the electrostatic potential of charge ${\displaystyle q_j}$ at ${\displaystyle {\mathbf{r}}_i}$. The total potential of the system is then the sum over ${\displaystyle j}$:

$${\displaystyle V=\frac{1}{8\pi {\epsilon }_0}\sum _{j=1}^N\sum _{i\ne j}\frac{q_i{q}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}}$$

so the Hamiltonian is:

$${\displaystyle \begin{array}{rl}\hat{H}& =-\frac{{\hslash }^2}{2}\sum _{j=1}^N\frac{1}{m_j}{\mathrm{\nabla }}_j^2+\frac{1}{8\pi {\epsilon }_0}\sum _{j=1}^N\sum _{i\ne j}\frac{q_i{q}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}\\ & =\sum _{j=1}^N\left(-\frac{{\hslash }^2}{2m_j}{\mathrm{\nabla }}_j^2+\frac{1}{8\pi {\epsilon }_0}\sum _{i\ne j}\frac{q_i{q}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j|}\right)\end{array}}$$

Electric dipole in an electric field

For an electric dipole moment ${\displaystyle \mathbf{d}}$ constituting charges of magnitude ${\displaystyle q}$, in a uniform, electrostatic field (time-independent) ${\displaystyle \mathbf{E}}$, positioned in one place, the potential is:

$${\displaystyle V=-\hat{\mathbf{d}}\cdot \mathbf{E}}$$

the dipole moment itself is the operator

$${\displaystyle \hat{\mathbf{d}}=q\hat{\mathbf{r}}}$$

Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:

$${\displaystyle \hat{H}=-\hat{\mathbf{d}}\cdot \mathbf{E}=-q\hat{\mathbf{r}}\cdot \mathbf{E}}$$

Magnetic dipole in a magnetic field

For a magnetic dipole moment ${\displaystyle \mathit{\mu }}$ in a uniform, magnetostatic field (time-independent) ${\displaystyle \mathbf{B}}$, positioned in one place, the potential is:

$${\displaystyle V=-\mathit{\mu }\cdot \mathbf{B}}$$

Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:

$${\displaystyle \hat{H}=-\mathit{\mu }\cdot \mathbf{B}}$$

For a spin-1⁄2 particle, the corresponding spin magnetic moment is:

$${\displaystyle {\mathit{\mu }}_S=\frac{g_{s}e}{2m}\mathbf{S}}$$

where ${\displaystyle g_s}$ is the "spin g-factor" (not to be confused with the gyromagnetic ratio), ${\displaystyle e}$ is the electron charge, ${\displaystyle \mathbf{S}}$ is the spin operator vector, whose components are the Pauli matrices, hence

$${\displaystyle \hat{H}=\frac{g_{s}e}{2m}\mathbf{S}\cdot \mathbf{B}}$$

Charged particle in an electromagnetic field

For a particle with mass ${\displaystyle m}$ and charge ${\displaystyle q}$ in an electromagnetic field, described by the scalar potential ${\displaystyle \varphi }$ and vector potential ${\displaystyle \mathbf{A}}$, there are two parts to the Hamiltonian to substitute for. The canonical momentum operator ${\displaystyle \hat{\mathbf{p}}}$, which includes a contribution from the ${\displaystyle \mathbf{A}}$ field and fulfils the canonical commutation relation, must be quantized;

$${\displaystyle \hat{\mathbf{p}}=m\dot{\mathbf{r}}+q\mathbf{A},}$$

where ${\displaystyle m\dot{\mathbf{r}}}$ is the kinetic momentum. The quantization prescription reads

$${\displaystyle \hat{\mathbf{p}}=-i\hslash \mathrm{\nabla },}$$

so the corresponding kinetic energy operator is

$${\displaystyle \hat{T}=\frac{1}{2}m\dot{\mathbf{r}}\cdot \dot{\mathbf{r}}=\frac{1}{2m}{\left(\hat{\mathbf{p}}-q\mathbf{A}\right)}^2}$$

and the potential energy, which is due to the ${\displaystyle \varphi }$ field, is given by

$${\displaystyle \hat{V}=q\varphi .}$$

Casting all of these into the Hamiltonian gives

$${\displaystyle \hat{H}=\frac{1}{2m}{\left(-i\hslash \mathrm{\nabla }-q\mathbf{A}\right)}^2+q\varphi .}$$

Energy eigenket degeneracy, symmetry, and conservation laws

In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the ${\displaystyle x}$ direction is a different state from one propagating in the ${\displaystyle y}$ direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.

It turns out that degeneracy occurs whenever a nontrivial unitary operator ${\displaystyle U}$ commutes with the Hamiltonian. To see this, suppose that ${\displaystyle |a⟩}$ is an energy eigenket. Then ${\displaystyle U|a⟩}$ is an energy eigenket with the same eigenvalue, since

$${\displaystyle UH|a⟩=U{E}_a|a⟩=E_a(U|a⟩)=H(U|a⟩).}$$

Since ${\displaystyle U}$ is nontrivial, at least one pair of ${\displaystyle |a⟩}$ and ${\displaystyle U|a⟩}$ must represent distinct states. Therefore, ${\displaystyle H}$ has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.

The existence of a symmetry operator implies the existence of a conserved observable. Let ${\displaystyle G}$ be the Hermitian generator of ${\displaystyle U}$:

$${\displaystyle U=I-i\epsilon G+O({\epsilon }^2)}$$

It is straightforward to show that if ${\displaystyle U}$ commutes with ${\displaystyle H}$, then so does ${\displaystyle G}$:

$${\displaystyle [H,G]=0}$$

Therefore,

$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}⟨\psi (t)|G|\psi (t)⟩=\frac{1}{i\hslash }⟨\psi (t)|[G,H]|\psi (t)⟩=0.}$$

In obtaining this result, we have used the Schrödinger equation, as well as its dual,

$${\displaystyle ⟨\psi (t)|H=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}⟨\psi (t)|.}$$

Thus, the expected value of the observable ${\displaystyle G}$ is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum.

Hamilton's equations

Hamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states ${\displaystyle \left\{|n⟩\right\}}$, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,

$${\displaystyle ⟨n^{\prime }|n⟩={\delta }_{n{n}^{\prime }}}$$

Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.

The instantaneous state of the system at time ${\displaystyle t}$, ${\displaystyle |\psi \left(t\right)⟩}$, can be expanded in terms of these basis states:

$${\displaystyle |\psi (t)⟩=\sum _n{a}_n(t)|n⟩}$$

where

$${\displaystyle a_n(t)=⟨n|\psi (t)⟩.}$$

The coefficients ${\displaystyle a_n(t)}$ are complex variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.

The expectation value of the Hamiltonian of this state, which is also the mean energy, is

$${\displaystyle ⟨H(t)⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}⟨\psi (t)|H|\psi (t)⟩=\sum _{n{n}^{\prime }}{a}_{n^{\prime }}^{\ast }{a}_n⟨n^{\prime }|H|n⟩}$$

where the last step was obtained by expanding ${\displaystyle |\psi \left(t\right)⟩}$ in terms of the basis states.

Each ${\displaystyle a_n(t)}$ actually corresponds to two independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use ${\displaystyle a_n(t)}$ and its complex conjugate ${\displaystyle a_n^{\ast }(t)}$. With this choice of independent variables, we can calculate the partial derivative

$${\displaystyle \frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{a}_{n^{\prime }}^{\ast }}=\sum _n{a}_n⟨n^{\prime }|H|n⟩=⟨n^{\prime }|H|\psi ⟩}$$

By applying Schrödinger's equation and using the orthonormality of the basis states, this further reduces to

$${\displaystyle \frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{a}_{n^{\prime }}^{\ast }}=i\hslash \frac{\mathrm{\partial }{a}_{n^{\prime }}}{\mathrm{\partial }t}}$$

Similarly, one can show that

$${\displaystyle \frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{a}_n}=-i\hslash \frac{\mathrm{\partial }{a}_n^{\ast }}{\mathrm{\partial }t}}$$

If we define "conjugate momentum" variables ${\displaystyle {\pi }_n}$ by

$${\displaystyle {\pi }_n(t)=i\hslash a_n^{\ast }(t)}$$

then the above equations become

$${\displaystyle \frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{\pi }_n}=\frac{\mathrm{\partial }{a}_n}{\mathrm{\partial }t},\frac{\mathrm{\partial }⟨H⟩}{\mathrm{\partial }{a}_n}=-\frac{\mathrm{\partial }{\pi }_n}{\mathrm{\partial }t}}$$

which is precisely the form of Hamilton's equations, with the ${\displaystyle a_n}$s as the generalized coordinates, the ${\displaystyle {\pi }_n}$s as the conjugate momenta, and ${\displaystyle ⟨H⟩}$ taking the place of the classical Hamiltonian.

Silicone

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Silicone

 

Silicone caulk can be used as a basic sealant against water and air penetration.

A silicone or polysiloxane is a polymer made up of siloxane (−R₂Si−O−SiR₂−, where R = organic group). They are typically colorless oils or rubber-like substances. Silicones are used in sealants, adhesives, lubricants, medicine, cooking utensils, thermal insulation, and electrical insulation. Some common forms include silicone oil, silicone grease, silicone rubber, silicone resin, and silicone caulk.

Chemistry

Main article: Organosilicon compound

 

Chemical structure of the silicone polydimethylsiloxane (PDMS)

Stock and Somieski examined the hydrolysis of dichlorosilane, a reaction that was proposed to initially give the monomer H₂SiO:

SiH₂Cl₂ + H₂O → H₂SiO + 2 HCl

When the hydrolysis is performed by treating a solution of H₂SiCl₂ in benzene with water, and the product was determined to have the approximate formula [H₂SiO]₆. Higher polymers were proposed to form with time.

Most polysiloxanes feature organic substituents, e.g., [(CH₃)₂SiO]_n and [(C₆H₅)₂SiO)]_n. All polymerized siloxanes or polysiloxanes, silicones consist of an inorganic silicon–oxygen backbone chain (⋯−Si−O−Si−O−Si−O−⋯) with two groups attached to each silicon center. The materials can be cyclic or polymeric. By varying the −Si−O− chain lengths, side groups, and crosslinking, silicones can be synthesized with a wide variety of properties and compositions. They can vary in consistency from liquid to gel to rubber to hard plastic. The most common siloxane is linear polydimethylsiloxane (PDMS), a silicone oil. The second-largest group of silicone materials is based on silicone resins, which are formed by branched and cage-like oligosiloxanes.

Terminology and history

F. S. Kipping coined the word silicone in 1901 to describe the formula of polydiphenylsiloxane, Ph₂SiO (Ph denoting phenyl, C₆H₅), by analogy with the formula of the ketone benzophenone, Ph₂CO (his term was originally silicoketone). Kipping was well aware that polydiphenylsiloxane is polymeric whereas benzophenone is monomeric and noted the contrasting properties of Ph₂SiO and Ph₂CO. The discovery of the structural differences between Kipping's molecules and the ketones means that silicone is no longer the correct term (though it remains in common usage) and that the term siloxane is preferred according to the nomenclature of modern chemistry.

James Franklin Hyde (born 11 March 1903) was an American chemist and inventor. He has been called the “Father of Silicones” and is credited with the launch of the silicone industry in the 1930s. His most notable contributions include his creation of silicone from silicon compounds and his method of making fused silica, a high-quality glass later used in aeronautics, advanced telecommunications, and computer chips. His work led to the formation of Dow Corning, an alliance between the Dow Chemical Company and Corning Glass Works that was specifically created to produce silicone products.

Silicone is often confused with silicon, but they are distinct substances. Silicon is a chemical element, a hard dark-grey semiconducting metalloid, which in its crystalline form is used to make integrated circuits ("electronic chips") and solar cells. Silicones are compounds that contain silicon, carbon, hydrogen, oxygen, and perhaps other kinds of atoms as well, and have many very different physical and chemical properties.

Compounds containing silicon–oxygen double bonds, now called silanones, but which could deserve the name "silicone", have long been identified as intermediates in gas-phase processes such as chemical vapor deposition in microelectronics production, and in the formation of ceramics by combustion. However, they have a strong tendency to polymerize into siloxanes. The first stable silanone was obtained in 2014 by A. Filippou and others.

Synthesis

Most common are materials based on polydimethylsiloxane, which is derived by hydrolysis of dimethyldichlorosilane. This dichloride reacts with water as follows:

n Si(CH₃)₂Cl₂ + n H₂O → [Si(CH₃)₂O]_n + 2n HCl

The polymerization typically produces linear chains capped with Si−Cl or Si−OH (silanol) groups. Under different conditions, the polymer is a cyclic, not a chain.

For consumer applications such as caulks silyl acetates are used instead of silyl chlorides. The hydrolysis of the acetates produces the less dangerous acetic acid (the acid found in vinegar) as the reaction product of a much slower curing process. This chemistry is used in many consumer applications, such as silicone caulk and adhesives.

n Si(CH₃)₂(CH₃COO)₂ + n H₂O → [Si(CH₃)₂O]_n + 2n CH₃COOH

Branches or crosslinks in the polymer chain can be introduced by using organosilicone precursors with fewer alkyl groups, such as methyl trichlorosilane and methyltrimethoxysilane. Ideally, each molecule of such a compound becomes a branch point. This process can be used to produce hard silicone resins. Similarly, precursors with three methyl groups can be used to limit molecular weight, since each such molecule has only one reactive site and so forms the end of a siloxane chain.

Combustion

When silicone is burned in air or oxygen, it forms solid silica (silicon dioxide, SiO₂) as a white powder, char, and various gases. The readily dispersed powder is sometimes called silica fume. The pyrolysis of certain polysiloxanes under an inert atmosphere is a valuable pathway towards the production of amorphous silicon oxycarbide ceramics, also known as polymer derived ceramics. Polysiloxanes terminated with functional ligands such as vinyl, mercapto or acrylate groups have been cross linked to yield preceramic polymers, which can be photopolymerised for the additive manufacturing of polymer derived ceramics by stereolithography techniques.

Properties

This silicone rubber folding chessboard resists creasing and wrinkling.

Silicones exhibit many useful characteristics, including:

• Low thermal conductivity
• Low chemical reactivity
• Low toxicity
• Thermal stability (constancy of properties over a wide temperature range of −100 to 250 °C)
• The ability to repel water and form watertight seals.
• Does not stick to many substrates, but adheres very well to others, e.g. glass
• Does not support microbiological growth
• Resistance to creasing and wrinkling
• Resistance to oxygen, ozone, and ultraviolet (UV) light. This property has led to the widespread use of silicones in the construction industry (e.g. coatings, fire protection, glazing seals) and the automotive industry (external gaskets, external trim).
• Electrical insulation properties. Because silicone can be formulated to be electrically insulative or conductive, it is suitable for a wide range of electrical applications.
• High gas permeability: at room temperature (25 °C), the permeability of silicone rubber for such gases as oxygen is approximately 400 times that of butyl rubber, making silicone useful for medical applications in which increased aeration is desired. Conversely, silicone rubbers cannot be used where gas-tight seals are necessary such as seals for high-pressure gasses or high vacuum.

Silicone can be developed into rubber sheeting, where it has other properties, such as being FDA compliant. This extends the uses of silicone sheeting to industries that demand hygiene, for example, food and beverage, and pharmaceuticals.

Applications

Silicones are used in many products. Ullmann's Encyclopedia of Industrial Chemistry lists the following major categories of application: Electrical (e.g. insulation), electronics (e.g., coatings), household (e.g., sealants and cooking utensils), automobile (e.g. gaskets), airplane (e.g., seals), office machines (e.g. keyboard pads), medicine and dentistry (e.g. tooth impression molds), textiles and paper (e.g. coatings). For these applications, an estimated 400,000 tonnes of silicones were produced in 1991. Specific examples, both large and small are presented below.

Automotive

Silicone caulks and rubber components are often used in automotive applications.

In the automotive field, silicone grease is typically used as a lubricant for brake components since it is stable at high temperatures, is not water-soluble, and is far less likely than other lubricants to foul. DOT 5 brake fluids are based on liquid silicones.

Automotive spark plug wires are insulated by multiple layers of silicone to prevent sparks from jumping to adjacent wires, causing misfires. Silicone tubing is sometimes used in automotive intake systems (especially for engines with forced induction).

Sheet silicone is used to manufacture gaskets used in automotive engines, transmissions, and other applications.

Automotive body manufacturing plants and paint shops avoid silicones, as trace contamination may cause "fish eyes", which are small, circular craters which mar a smooth finish.

Additionally, silicone compounds such as silicone rubber are used as coatings and sealants for airbags; the high strength of silicone rubber makes it an optimal adhesive and sealant for high impact airbags. Silicones in combination with thermoplastics provide improvements in scratch and mar resistance and lowered coefficient of friction.

Aerospace

Silicone is often used to seal maintenance access openings in aerospace equipment.

Silicone is a widely used material in the aerospace industry due to its sealing properties, stability across an extreme temperature range, durability, sound dampening and anti-vibration qualities, and naturally flame retardant properties. Maintaining extreme functionality is paramount for passenger safety in the aerospace industry, so each component on an aircraft requires high-performance materials.

Specially developed aerospace grades of silicone are stable from −70 to 220 °C, these grades can be used in the construction of gaskets for windows and cabin doors. During operation, aircraft go through large temperature fluctuations in a relatively short period of time; from freezing temperatures when flying at full altitude to the ambient temperatures when on the ground in hot countries. Silicone rubber can be molded with tight tolerances ensuring gaskets form airtight seals both on the ground and in the air, where atmospheric pressure decreases.

Silicone rubber's resistance to heat corrosion enables it to be used for gaskets in aircraft engines where it will outlast other types of rubber, both improving aircraft safety and reducing maintenance costs. The silicone acts to seal instrument panels and other electrical systems in the cockpit, protecting printed circuit boards from the risks of extreme altitude such as moisture and extremely low temperature. Silicone can be used as a sheath to protect wires and electrical components from any dust or ice that may creep into a plane's inner workings.

As the nature of air travel results in much noise and vibration, powerful engines, landings, and high speeds all need to be considered to ensure passenger comfort and safe operation of the aircraft. As silicone rubber has exceptional noise reduction and anti-vibration properties, it can be formed into small components and fitted into small gaps ensuring all equipment can be protected from unwanted vibration such as overhead lockers, vent ducts, hatches, entertainment system seals, and LED lighting systems.

Solid propellant

Polydimethylsiloxane (PDMS) based binders along with ammonium perchlorate (NH₄ClO₄) are used as fast burning solid propellants in rockets.

Building construction

The strength and reliability of silicone rubber are widely acknowledged in the construction industry. One-part silicone sealants and caulks are in common use to seal gaps, joints and crevices in buildings. One-part silicones cure by absorbing atmospheric moisture, which simplifies installation. In plumbing, silicone grease is typically applied to O-rings in brass taps and valves, preventing lime from sticking to the metal.

Structural silicone has also been used in curtain wall building façades since 1974 when the Art Institute of Chicago became the first building to receive exterior glass fixed only with the material. Silicone membranes have been used to cover and restore industrial roofs, thanks to its extreme UV resistance, and ability to keep their waterproof performance for decades.

Coatings

Silicone films can be applied to such silica-based substrates as glass to form a covalently bonded hydrophobic coating. Such coatings were developed for use on aircraft windshields to repel water and to preserve visibility, without requiring mechanical windshield wipers which are impractical at supersonic speeds. Similar treatments were eventually adapted to the automotive market in products marketed by Rain-X and others.

Many fabrics can be coated or impregnated with silicone to form a strong, waterproof composite such as silnylon.

A silicone polymer can be suspended in water by using stabilizing surfactants. This allows water-based formulations to be used to deliver many ingredients that would otherwise require a stronger solvent, or be too viscous to use effectively. For example, a waterborne formulation using a silane's reactivity and penetration ability into a mineral-based surface can be combined with water-beading properties from a siloxane to produce a more-useful surface protection product.

Cookware

As a low-taint, non-toxic material, silicone can be used where contact with food is required. Silicone is becoming an important product in the cookware industry, particularly bakeware and kitchen utensils. Silicone is used as an insulator in heat-resistant potholders and similar items; however, it is more conductive of heat than similar less dense fiber-based products. Silicone oven mitts are able to withstand temperatures up to 260 °C (500 °F), making it possible to reach into boiling water.

Other products include molds for chocolate, ice, cookies, muffins, and various other foods; non-stick bakeware and reusable mats used on baking sheets; steamers, egg boilers or poachers; cookware lids, pot holders, trivets, and kitchen mats.

• 

  Soup ladle and pasta ladle made of silicone

• 

  A silicone food steamer to be placed inside a pot of boiling water

• 

  Flexible ice cube trays made of silicone allow easy extraction of ice.

• 

  Silicone brush used for basting and applying flavoring liquids

Defoaming

Silicones are used as active compounds in defoamers due to their low water solubility and good spreading properties.

Dry cleaning

Liquid silicone can be used as a dry cleaning solvent, providing an alternative to the traditional chlorine-containing perchloroethylene (perc) solvent. The use of silicones in dry cleaning reduces the environmental effect of a typically high-polluting industry.

Electronics

Silicone rubber keypad

Electronic components are sometimes encased in silicone to increase stability against mechanical and electrical shock, radiation and vibration, a process called "potting". Silicones are used where durability and high performance are demanded of components under extreme environmental conditions, such as in space (satellite technology). They are selected over polyurethane or epoxy encapsulation when a wide operating temperature range is required (−65 to 315 °C). Silicones also have the advantage of little exothermic heat rise during cure, low toxicity, good electrical properties, and high purity.

Silicones are often components of thermal pastes used to improve heat transfer from power-dissipating electronic components to heat sinks.

The use of silicones in electronics is not without problems, however. Silicones are relatively expensive and can be attacked by certain solvents. Silicone easily migrates as either a liquid or vapor onto other components. Silicone contamination of electrical switch contacts can lead to failures by causing an increase in contact resistance, often late in the life of the contact, well after any testing is completed. Use of silicone-based spray products in electronic devices during maintenance or repairs can cause later failures.

Firestops

Red-colored silicone firestopping

Silicone foam has been used in North American buildings in an attempt to firestop openings within the fire-resistance-rated wall and floor assemblies to prevent the spread of flames and smoke from one room to another. When properly installed, silicone-foam firestops can be fabricated for building code compliance. Advantages include flexibility and high dielectric strength. Disadvantages include combustibility (hard to extinguish) and significant smoke development.

Silicone-foam firestops have been the subject of controversy and press attention due to smoke development from pyrolysis of combustible components within the foam, hydrogen gas escape, shrinkage, and cracking. These problems have led to reportable events among licensees (operators of nuclear power plants) of the Nuclear Regulatory Commission (NRC).

Silicone firestops are also used in aircraft.

Jewelry

Silicone is a popular alternative to traditional metals (such as silver and gold) with jewelry, specifically rings. Silicone rings are commonly worn in professions where metal rings can lead to injuries, such as electrical conduction and ring avulsions. During the mid-2010's, some professional athletes began wearing silicone rings as an alternative during games.

Lubricants

Silicone grease is often used with laboratory glassware to prevent seizing.

Silicone greases are used for many purposes, such as bicycle chains, airsoft gun parts, and a wide range of other mechanisms. Typically, a dry-set lubricant is delivered with a solvent carrier to penetrate the mechanism. The solvent then evaporates, leaving a clear film that lubricates but does not attract dirt and grit as much as an oil-based or other traditional "wet" lubricant.

Silicone personal lubricants are also available for use in medical procedures or sexual activity.

Medicine and cosmetic surgery

Silicone is used in microfluidics, seals, gaskets, shrouds, and other applications requiring high biocompatibility. Additionally, the gel form is used in bandages and dressings, breast implants, testicle implants, pectoral implants, contact lenses, and a variety of other medical uses.

Scar treatment sheets are often made of medical grade silicone due to its durability and biocompatibility. Polydimethylsiloxane (PDMS) is often used for this purpose, since its specific crosslinking results in a flexible and soft silicone with high durability and tack. It has also been used as the hydrophobic block of amphiphilic synthetic block copolymers used to form the vesicle membrane of polymersomes.

Illicit cosmetic silicone injections may induce chronic and definitive silicone blood diffusion with dermatologic complications.

Ophthalmology uses many products such as silicone oil used to replace the vitreous humor following vitrectomy, silicone intraocular lenses following cataract extraction, silicone tubes to keep a nasolacrimal passage open following dacryocystorhinostomy, canalicular stents for canalicular stenosis, punctual plugs for punctual occlusion in dry eyes, silicone rubber and bands as an external tamponade in tractional retinal detachment, and anteriorly-located break in rhegmatogenous retinal detachment.

Addition and condensation (e.g. polyvinyl siloxane) silicones find wide application as a dental impression material due to its hydrophobic property and thermal stability.

Moldmaking

Silicone mold used to reproduce an architectural detail

Two-part silicone systems are used as rubber molds to cast resins, foams, rubber, and low-temperature alloys. A silicone mold generally requires little or no mold-release or surface preparation, as most materials do not adhere to silicone. For experimental uses, ordinary one-part silicone can be used to make molds or to mold into shapes. If needed, common vegetable cooking oils or petroleum jelly can be used on mating surfaces as a mold-release agent.

Silicone cooking molds used as bakeware do not require coating with cooking oil; in addition, the flexibility of the rubber allows the baked food to be easily removed from the mold after cooking.

Personal care

Silicone rubber earplugs for hearing protection

Silicones are ingredients widely used in skincare, color cosmetic and hair care applications. Some silicones, notably the amine functionalized amodimethicones, are excellent hair conditioners, providing improved compatibility, feel, and softness, and lessening frizz. The phenyl dimethicones, in another silicone family, are used in reflection-enhancing and color-correcting hair products, where they increase shine and glossiness (and possibly impart subtle color changes). Phenyltrimethicones, unlike the conditioning amodimethicones, have refractive indices (typically 1.46) close to that of a human hair (1.54). However, if included in the same formulation, amodimethicone and phenyltrimethicone interact and dilute each other, making it difficult to achieve both high shine and excellent conditioning in the same product.

Silicone rubber is commonly used in baby bottle nipples (teats) for its cleanliness, aesthetic appearance, and low extractable content.

Silicones are used in shaving products and personal lubricants.

Toys and hobbies

Baby toys made of nontoxic silicone rubber

Silly Putty and similar materials are composed of silicones dimethyl siloxane, polydimethylsiloxane, and decamethyl cyclopentasiloxane, with other ingredients. This substance is noted for its unusual characteristics, e.g., that it bounces, but breaks when given a sharp blow; it will also flow like a liquid and form a puddle given enough time.

Silicone "rubber bands" are a long-lasting popular replacement refill for real rubber bands in the 2013 fad "rubber band loom" toys at two to four times the price (in 2014). Silicone bands also come in bracelet sizes that can be custom embossed with a name or message. Large silicone bands are also sold as utility tie-downs.

Formerol is a silicone rubber (marketed as Sugru) used as an arts-and-crafts material, as its plasticity allows it to be molded by hand like modeling clay. It hardens at room temperature and it is adhesive to various substances including glass and aluminum.

Oogoo is an inexpensive silicone clay, which can be used as a substitute for Sugru.

In making aquariums, manufacturers now commonly use 100% silicone sealant to join glass plates. Glass joints made with silicone sealant can withstand great pressure, making obsolete the original aquarium construction method of angle-iron and putty. This same silicone is used to make hinges in aquarium lids or for minor repairs. However, not all commercial silicones are safe for aquarium manufacture, nor is silicone used for the manufacture of acrylic aquariums as silicones do not have long-term adhesion to plastics.

Production and marketing

The leading global manufacturers of silicone base materials belong to three regional organizations: the European Silicone Center (CES) in Brussels, Belgium; the Environment Health and Safety Council (SEHSC) in Herndon, Virginia, US; and the Silicone Industry Association of Japan (SIAJ) in Tokyo, Japan. Dow Corning Silicones, Evonik Industries, Momentive Performance Materials, Milliken and Company (SiVance Specialty Silicones), Shin-Etsu Silicones, Wacker Chemie, Bluestar Silicones, JNC Corporation, Wacker Asahikasei Silicone, and Dow Corning Toray represent the collective membership of these organizations. A fourth organization, the Global Silicone Council (GSC) acts as an umbrella structure over the regional organizations. All four are non-profit, having no commercial role; their primary missions are to promote the safety of silicones from a health, safety, and environmental perspective. As the European chemical industry is preparing to implement the Registration, Evaluation, and Authorisation of Chemicals (REACH) legislation, CES is leading the formation of a consortium of silicones, silanes, and siloxanes producers and importers to facilitate data and cost-sharing.

Safety and environmental considerations

Silicone compounds are pervasive in the environment. Particular silicone compounds, cyclic siloxanes D₄ and D₅, are air and water pollutants and have negative health effects on test animals. They are used in various personal care products. The European Chemicals Agency found that "D₄ is a persistent, bioaccumulative and toxic (PBT) substance and D₅ is a very persistent, very bioaccumulative (vPvB) substance". Other silicones biodegrade readily, a process that is accelerated by a variety of catalysts, including clays. Cyclic silicones have been shown to involve the occurrence of silanols during biodegradation in mammals. The resulting silanediols and silanetriols are capable of inhibiting hydrolytic enzymes such as thermolysin, acetycholinesterase. However, the doses required for inhibition are by orders of magnitude higher than the ones resulting from the accumulated exposure to consumer products containing cyclomethicone.

At around 200 °C (392 °F) in an oxygen-containing atmosphere, PDMS releases traces of formaldehyde (but lesser amounts than other common materials such as polyethylene.) At this temperature, silicones were found to have lower formaldehyde generation than mineral oil and plastics (less than 3 to 48 µg CH₂O/(g·hr) for a high consistency silicone rubber, versus around 400 µg CH₂O/(g·hr) for plastics and mineral oil). By 250 °C (482 °F), copious amounts of formaldehyde have been found to be produced by all silicones (1,200 to 4,600 µg CH₂O/(g·hr)).

Polycarbonate

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Polycarbonate

Polycarbonates (PC) are a group of thermoplastic polymers containing carbonate groups in their chemical structures. Polycarbonates used in engineering are strong, tough materials, and some grades are optically transparent. They are easily worked, molded, and thermoformed. Because of these properties, polycarbonates find many applications. Polycarbonates do not have a unique resin identification code (RIC) and are identified as "Other", 7 on the RIC list. Products made from polycarbonate can contain the precursor monomer bisphenol A (BPA).

Structure

Carbonate esters have planar OC(OC)₂ cores, which confers rigidity. The unique O=C bond is short (1.173 Å in the depicted example), while the C-O bonds are more ether-like (the bond distances of 1.326 Å for the example depicted). Polycarbonates received their name because they are polymers containing carbonate groups (−O−(C=O)−O−). A balance of useful features, including temperature resistance, impact resistance and optical properties, positions polycarbonates between commodity plastics and engineering plastics.

Production

Phosgene route

The main polycarbonate material is produced by the reaction of bisphenol A (BPA) and phosgene COCl₂. The overall reaction can be written as follows:

The first step of the synthesis involves treatment of bisphenol A with sodium hydroxide, which deprotonates the hydroxyl groups of the bisphenol A.

(HOC₆H₄)₂CMe₂ + 2 NaOH → Na₂(OC₆H₄)₂CMe₂ + 2 H₂O

The diphenoxide (Na₂(OC₆H₄)₂CMe₂) reacts with phosgene to give a chloroformate, which subsequently is attacked by another phenoxide. The net reaction from the diphenoxide is:

Na₂(OC₆H₄)₂CMe₂ + COCl₂ → 1/n [OC(OC₆H₄)₂CMe₂]_n + 2 NaCl

In this way, approximately one billion kilograms of polycarbonate is produced annually. Many other diols have been tested in place of bisphenol A, e.g. 1,1-bis(4-hydroxyphenyl)cyclohexane and dihydroxybenzophenone. The cyclohexane is used as a comonomer to suppress crystallisation tendency of the BPA-derived product. Tetrabromobisphenol A is used to enhance fire resistance. Tetramethylcyclobutanediol has been developed as a replacement for BPA.

Transesterification route

An alternative route to polycarbonates entails transesterification from BPA and diphenyl carbonate:

(HOC₆H₄)₂CMe₂ + (C₆H₅O)₂CO → 1/n [OC(OC₆H₄)₂CMe₂]_n + 2 C₆H₅OH

Properties and processing

Polycarbonate is a durable material. Although it has high impact-resistance, it has low scratch-resistance. Therefore, a hard coating is applied to polycarbonate eyewear lenses and polycarbonate exterior automotive components. The characteristics of polycarbonate compare to those of polymethyl methacrylate (PMMA, acrylic), but polycarbonate is stronger and will hold up longer to extreme temperature. Thermally processed material is usually totally amorphous, and as a result is highly transparent to visible light, with better light transmission than many kinds of glass.

Polycarbonate has a glass transition temperature of about 147 °C (297 °F), so it softens gradually above this point and flows above about 155 °C (311 °F).ools must be held at high temperatures, generally above 80 °C (176 °F) to make strain-free and stress-free products. Low molecular mass grades are easier to mold than higher grades, but their strength is lower as a result. The toughest grades have the highest molecular mass, but are more difficult to process.

Unlike most thermoplastics, polycarbonate can undergo large plastic deformations without cracking or breaking. As a result, it can be processed and formed at room temperature using sheet metal techniques, such as bending on a brake. Even for sharp angle bends with a tight radius, heating may not be necessary. This makes it valuable in prototyping applications where transparent or electrically non-conductive parts are needed, which cannot be made from sheet metal. PMMA/Acrylic, which is similar in appearance to polycarbonate, is brittle and cannot be bent at room temperature.

Main transformation techniques for polycarbonate resins:

• extrusion into tubes, rods and other profiles including multiwall
• extrusion with cylinders (calenders) into sheets (0.5–20 mm (0.020–0.787 in)) and films (below 1 mm (0.039 in)), which can be used directly or manufactured into other shapes using thermoforming or secondary fabrication techniques, such as bending, drilling, or routing. Due to its chemical properties it is not conducive to laser-cutting.
• injection molding into ready articles

Polycarbonate may become brittle when exposed to ionizing radiation above 25 kGy (kJ/kg).

A bottle made from polycarbonate

Applications

Electronic components

Polycarbonate is mainly used for electronic applications that capitalize on its collective safety features. A good electrical insulator with heat-resistant and flame-retardant properties, it is used in products associated with power systems and telecommunications hardware. It can serve as a dielectric in high-stability capacitors. Commercial manufacture of polycarbonate capacitors mostly stopped after sole manufacturer Bayer AG stopped making capacitor-grade polycarbonate film at the end of 2000.

Construction materials

Polycarbonate sheeting in a greenhouse

The second largest consumer of polycarbonates is the construction industry, e.g. for domelights, flat or curved glazing, roofing sheets and sound walls. Polycarbonates are used to create materials used in buildings that must be durable but light.

3D Printing

Polycarbonates are used extensively in 3D FDM printing, producing durable strong plastic products with a high melting point. Polycarbonate is relatively difficult for casual hobbyists to print compared to thermoplastics such as Polylactic acid (PLA) or Acrylonitrile butadiene styrene (ABS) because of the high melting point, difficulty with print bed adhesion, tendency to warp during printing, and tendency to absorb moisture in humid environments. Despite these issues, 3D printing using polycarbonates is common in the professional community.

Data storage

CDs and DVDs

A major polycarbonate market is the production of compact discs, DVDs, and Blu-ray discs. These discs are produced by injection-molding polycarbonate into a mold cavity that has on one side a metal stamper containing a negative image of the disc data, while the other mold side is a mirrored surface. Typical products of sheet/film production include applications in advertisement (signs, displays, poster protection).

Automotive, aircraft, and security components

In the automotive industry, injection-molded polycarbonate can produce very smooth surfaces that make it well-suited for sputter deposition or evaporation deposition of aluminium without the need for a base-coat. Decorative bezels and optical reflectors are commonly made of polycarbonate. Its low weight and high impact resistance have made polycarbonate the dominant material for automotive headlamp lenses. However, automotive headlamps require outer surface coatings because of its low scratch resistance and susceptibility to ultraviolet degradation (yellowing). The use of polycarbonate in automotive applications is limited to low stress applications. Stress from fasteners, plastic welding and molding render polycarbonate susceptible to stress corrosion cracking when it comes in contact with certain accelerants such as salt water and plastisol. It can be laminated to make bullet-proof "glass", although "bullet-resistant" is more accurate for the thinner windows, such as are used in bullet-resistant windows in automobiles. The thicker barriers of transparent plastic used in teller's windows and barriers in banks are also polycarbonate.

So-called "theft-proof" large plastic packaging for smaller items, which cannot be opened by hand, is typically made from polycarbonate.

Lockheed Martin F-22 cockpit canopy

The cockpit canopy of the Lockheed Martin F-22 Raptor jet fighter is fabricated from high optical quality polycarbonate. It is the largest item of its type.

Niche applications

Polycarbonate, being a versatile material with attractive processing and physical properties, has attracted myriad smaller applications. The use of injection molded drinking bottles, glasses and food containers is common, but the use of BPA in the manufacture of polycarbonate has stirred concerns (see Potential hazards in food contact applications), leading to development and use of "BPA-free" plastics in various formulations.

Laboratory safety goggles

Polycarbonate is commonly used in eye protection, as well as in other projectile-resistant viewing and lighting applications that would normally indicate the use of glass, but require much higher impact-resistance. Polycarbonate lenses also protect the eye from UV light. Many kinds of lenses are manufactured from polycarbonate, including automotive headlamp lenses, lighting lenses, sunglass/eyeglass lenses, swimming goggles and SCUBA masks, and safety glasses/goggles/visors including visors in sporting helmets/masks and police riot gear (helmet visors, riot shields, etc.). Windscreens in small motorized vehicles are commonly made of polycarbonate, such as for motorcycles, ATVs, golf carts, and small airplanes and helicopters.

The light weight of polycarbonate as opposed to glass has led to development of electronic display screens that replace glass with polycarbonate, for use in mobile and portable devices. Such displays include newer e-ink and some LCD screens, though CRT, plasma screen and other LCD technologies generally still require glass for its higher melting temperature and its ability to be etched in finer detail.

As more and more governments are restricting the use of glass in pubs and clubs due to the increased incidence of glassings, polycarbonate glasses are becoming popular for serving alcohol because of their strength, durability, and glass-like feel.

Other miscellaneous items include durable, lightweight luggage, MP3/digital audio player cases, ocarinas, computer cases, riot shields, instrument panels, tealight candle containers and food blender jars. Many toys and hobby items are made from polycarbonate parts, like fins, gyro mounts, and flybar locks in radio-controlled helicopters, and transparent LEGO (ABS is used for opaque pieces).

Standard polycarbonate resins are not suitable for long term exposure to UV radiation. To overcome this, the primary resin can have UV stabilisers added. These grades are sold as UV stabilized polycarbonate to injection moulding and extrusion companies. Other applications, including polycarbonate sheets, may have the anti-UV layer added as a special coating or a coextrusion for enhanced weathering resistance.

Polycarbonate is also used as a printing substrate for nameplate and other forms of industrial grade under printed products. The polycarbonate provides a barrier to wear, the elements, and fading.

Medical applications

Many polycarbonate grades are used in medical applications and comply with both ISO 10993-1 and USP Class VI standards (occasionally referred to as PC-ISO). Class VI is the most stringent of the six USP ratings. These grades can be sterilized using steam at 120 °C, gamma radiation, or by the ethylene oxide (EtO) method. Trinseo strictly limits all its plastics with regard to medical applications. Aliphatic polycarbonates have been developed with improved biocompatibility and degradability for nanomedicine applications.

Mobile phones

Some smartphone manufacturers use polycarbonate. Nokia used polycarbonate in their phones starting with the N9's unibody case in 2011. This practice continued with various phones in the Lumia series. Samsung started using polycarbonate with Galaxy S III's hyperglaze-branded removable battery cover in 2012. This practice continues with various phones in the Galaxy series. Apple started using polycarbonate with the iPhone 5C's unibody case in 2013.

Benefits over glass and metal back covers include durability against shattering (advantage over glass), bending and scratching (advantage over metal), shock absorption, low manufacturing costs, and no interference with radio signals and wireless charging (advantage over metal). Polycarbonate back covers are available in glossy or matte surface textures.

History

Polycarbonates were first discovered in 1898 by Alfred Einhorn, a German scientist working at the University of Munich. However, after 30 years' laboratory research, this class of materials was abandoned without commercialization. Research resumed in 1953, when Hermann Schnell at Bayer in Uerdingen, Germany patented the first linear polycarbonate. The brand name "Makrolon" was registered in 1955.

Also in 1953, and one week after the invention at Bayer, Daniel Fox at General Electric (GE) in Pittsfield, Massachusetts, independently synthesized a branched polycarbonate. Both companies filed for U.S. patents in 1955, and agreed that the company lacking priority would be granted a license to the technology.

Patent priority was resolved in Bayer's favor, and Bayer began commercial production under the trade name Makrolon in 1958. GE began production under the name Lexan in 1960, creating the GE Plastics division in 1973.

After 1970, the original brownish polycarbonate tint was improved to "glass-clear".

Potential hazards in food contact applications

Main articles: Bisphenol A and Endocrine disruptor

The use of polycarbonate containers for the purpose of food storage is controversial. The basis of this controversy is their hydrolysis (degradation by water, often referred to as leaching) occurring at high temperature, releases bisphenol A:

1/n [OC(OC₆H₄)₂CMe₂]_n + H₂O → (HOC₆H₄)₂CMe₂ + CO₂

More than 100 studies have explored the bioactivity of bisphenol A derived from polycarbonates. Bisphenol A appeared to be released from polycarbonate animal cages into water at room temperature and it may have been responsible for enlargement of the reproductive organs of female mice. However, the animal cages used in the research were fabricated from industrial grade polycarbonate, rather than FDA food grade polycarbonate.

An analysis of the literature on bisphenol A leachate low-dose effects by vom Saal and Hughes published in August 2005 seems to have found a suggestive correlation between the source of funding and the conclusion drawn. Industry-funded studies tend to find no significant effects whereas government-funded studies tend to find significant effects.

Sodium hypochlorite bleach and other alkali cleaners catalyze the release of the bisphenol A from polycarbonate containers. Polycarbonate is incompatible with ammonia and acetone. Alcohol is a recommended organic solvent for cleaning grease and oils from polycarbonate.

Environmental impact

Disposal

Studies have shown that at temperatures above 70 °C, and high humidity, polycarbonate will hydrolyze to bisphenol A (BPA). After about 30 days at 85 °C/96% RH, surface crystals are formed which for 70% consisted of BPA. BPA is a compound that is currently on the list of potential environmental hazardous chemicals. It is on the watch list of many countries, such as United States and Germany.

-(-OC₆H₄)₂C(CH₃)₂CO-)-_n + H₂O → (CH₃)₂C(C₆H₄OH)₂ + CO₂

The leaching of BPA from polycarbonate can also occur at environmental temperature and normal pH (in landfills).The amount of leaching increases as the polycarbonate parts get older. A study found that the decomposition of BPA in landfills (under anaerobic conditions) will not occur. It will therefore be persistent in landfills. Eventually, it will find its way into water bodies and contribute to aquatic pollution.

Photo-oxidation of polycarbonate

In the presence of UV light, oxidation of this polymer yields compounds such as ketones, phenols, o-phenoxybenzoic acid, benzyl alcohol and other unsaturated compounds. This has been suggested through kinetic and spectral studies. The yellow color formed after long exposure to sun can also be related to further oxidation of phenolic end group.

(OC₆H₄)₂C(CH₃)₂CO )_n + O₂ , R* → (OC₆H₄)₂C(CH₃CH₂)CO)_n

This product can be further oxidized to form smaller unsaturated compounds. This can proceed via two different pathways, the products formed depends on which mechanism takes place.

Pathway A

(OC₆H₄)₂C(CH₃CH₂)CO + O₂, H* ${\displaystyle ⟶}$ HO(OC₆H₄)OCO + CH₃COCH₂(OC₆H₄)OCO

Pathway B

(OC₆H₄)₂C(CH₃CH₂)CO)_n + O₂, H* ${\displaystyle ⟶}$ OCO(OC₆H₄)CH₂OH + OCO(OC₆H₄)COCH₃

Photo-aging reaction

Photo-aging is another degradation route for polycarbonates. Polycarbonate molecules (such as the aromatic ring) absorb UV radiation. This absorbed energy causes cleavage of covalent bonds which initiates the photo-aging process. The reaction can be propagated via side chain oxidation, ring oxidation or photo-Fries rearrangement. Products formed include phenyl salicylate, dihydroxybenzophenone groups, and hydroxydiphenyl ether groups.

(C₁₆H₁₄O₃)_n ${\displaystyle ⟶}$ C₁₆H₁₇O₃ + C₁₃H₁₀O₃

Thermal degradation

Waste polycarbonate will degrade at high temperatures to form solid, liquid and gaseous pollutants. A study showed that the products were about 40–50 wt.% liquid, 14–16 wt.% gases, while 34–43 wt.% remained as solid residue. Liquid products contained mainly phenol derivatives (∼75wt.%) and bisphenol (∼10wt.%) also present. Polycarbonate, however, can be safely used as a carbon source in the steel-making industry.

Phenol derivatives are environmental pollutants, classified as volatile organic compounds (VOC). Studies show they are likely to facilitate ground level ozone formation and increase photo-chemical smog. In aquatic bodies, they can potentially accumulate in organisms. They are persistent in landfills, do not readily evaporate and would remain in the atmosphere.

Effect of fungi

In 2001 a species of fungus in Belize, Geotrichum candidum, was found to consume the polycarbonate found in compact discs (CD). This has prospects for bioremediation. However, this effect has not been reproduced.