History of general relativity

From Wikipedia, the free encyclopedia

General relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915, with contributions by many others after 1915. According to general relativity, the observed gravitational attraction between masses results from the warping of space and time by those masses.

Before the advent of general relativity, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses, even though Newton himself did not regard the theory as the final word on the nature of gravity. Within a century of Newton's formulation, careful astronomical observation revealed unexplainable variations between the theory and the observations. Under Newton's model, gravity was the result of an attractive force between massive objects. Although even Newton was bothered by the unknown nature of that force, the basic framework was extremely successful at describing motion.

However, experiments and observations show that Einstein's description accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment or observation, while others are the subject of ongoing research.

General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where gravitational attraction is so strong that not even light can escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.

Creation of general relativity

Early investigations

As Einstein later said, the reason for the development of general relativity was the preference of inertial motion within special relativity, while a theory which from the outset prefers no state of motion appeared more satisfactory to him. So, while still working at the patent office in 1907, Einstein had what he would call his "happiest thought". He realized that the principle of relativity could be extended to gravitational fields.

Consequently, in 1907 he wrote an article (published 1908) on acceleration under special relativity. In that article, he argued that free fall is really inertial motion, and that for a freefalling observer the rules of special relativity must apply. This argument is called the Equivalence principle. In the same article, Einstein also predicted the phenomenon of gravitational time dilation.

In 1911, Einstein published another article expanding on the 1907 article. There, he thought about the case of a uniformly accelerated box not in a gravitational field, and noted that it would be indistinguishable from a box sitting still in an unchanging gravitational field. He used special relativity to see that the rate of clocks at the top of a box accelerating upward would be faster than the rate of clocks at the bottom. He concludes that the rates of clocks depend on their position in a gravitational field, and that the difference in rate is proportional to the gravitational potential to first approximation.

Also the deflection of light by massive bodies was predicted. Although the approximation was crude, it allowed him to calculate that the deflection is nonzero. German astronomer Erwin Finlay-Freundlich publicized Einstein's challenge to scientists around the world. This urged astronomers to detect the deflection of light during a solar eclipse, and gave Einstein confidence that the scalar theory of gravity proposed by Gunnar Nordström was incorrect. But the actual value for the deflection that he calculated was too small by a factor of two, because the approximation he used doesn't work well for things moving at near the speed of light. When Einstein finished the full theory of general relativity, he would rectify this error and predict the correct amount of light deflection by the sun.

Another of Einstein's notable thought experiments about the nature of the gravitational field is that of the rotating disk (a variant of the Ehrenfest paradox). He imagined an observer performing experiments on a rotating turntable. He noted that such an observer would find a different value for the mathematical constant π than the one predicted by Euclidean geometry. The reason is that the radius of a circle would be measured with an uncontracted ruler, but, according to special relativity, the circumference would seem to be longer because the ruler would be contracted. Since Einstein believed that the laws of physics were local, described by local fields, he concluded from this that spacetime could be locally curved. This led him to study Riemannian geometry, and to formulate general relativity in this language.

Developing general relativity

Eddington's photograph of a solar eclipse, which confirmed Einstein's theory that light "bends".

In 1912, Einstein returned to Switzerland to accept a professorship at his alma mater, the ETH. Once back in Zurich, he immediately visited his old ETH classmate Marcel Grossmann, now a professor of mathematics, who introduced him to Riemannian geometry and, more generally, to differential geometry. On the recommendation of Italian mathematician Tullio Levi-Civita, Einstein began exploring the usefulness of general covariance (essentially the use of tensors) for his gravitational theory. For a while Einstein thought that there were problems with the approach, but he later returned to it and, by late 1915, had published his general theory of relativity in the form in which it is used today. This theory explains gravitation as distortion of the structure of spacetime by matter, affecting the inertial motion of other matter.

During World War I, the work of Central Powers scientists was available only to Central Powers academics, for national security reasons. Some of Einstein's work did reach the United Kingdom and the United States through the efforts of the Austrian Paul Ehrenfest and physicists in the Netherlands, especially 1902 Nobel Prize-winner Hendrik Lorentz and Willem de Sitter of Leiden University. After the war ended, Einstein maintained his relationship with Leiden University, accepting a contract as an Extraordinary Professor; for ten years, from 1920 to 1930, he travelled to the Netherlands regularly to lecture.

In 1917, several astronomers accepted Einstein's 1911 challenge from Prague. The Mount Wilson Observatory in California, U.S., published a solar spectroscopic analysis that showed no gravitational redshift. In 1918, the Lick Observatory, also in California, announced that it too had disproved Einstein's prediction, although its findings were not published.

However, in May 1919, a team led by the British astronomer Arthur Stanley Eddington claimed to have confirmed Einstein's prediction of gravitational deflection of starlight by the Sun while photographing a solar eclipse with dual expeditions in Sobral, northern Brazil, and Príncipe, a west African island. Nobel laureate Max Born praised general relativity as the "greatest feat of human thinking about nature"; fellow laureate Paul Dirac was quoted saying it was "probably the greatest scientific discovery ever made". The international media guaranteed Einstein's global renown.

There have been claims that scrutiny of the specific photographs taken on the Eddington expedition showed the experimental uncertainty to be comparable to the same magnitude as the effect Eddington claimed to have demonstrated, and that a 1962 British expedition concluded that the method was inherently unreliable. The deflection of light during a solar eclipse was confirmed by later, more accurate observations. Some resented the newcomer's fame, notably among some German physicists, who later started the Deutsche Physik (German Physics) movement.

General covariance and the hole argument

By 1912, Einstein was actively seeking a theory in which gravitation was explained as a geometric phenomenon. At the urging of Tullio Levi-Civita, Einstein began by exploring the use of general covariance (which is essentially the use of curvature tensors) to create a gravitational theory. However, in 1913 Einstein abandoned that approach, arguing that it is inconsistent based on the "hole argument". In 1914 and much of 1915, Einstein was trying to create field equations based on another approach. When that approach was proven to be inconsistent, Einstein revisited the concept of general covariance and discovered that the hole argument was flawed.

The development of the Einstein field equations

When Einstein realized that general covariance was actually tenable, he quickly completed the development of the field equations that are named after him. However, he made a now-famous mistake. The field equations he published in October 1915 were

${\displaystyle R_{\mu \nu }=T_{\mu \nu }\,}$,

where ${\displaystyle R_{\mu \nu }}$ is the Ricci tensor, and ${\displaystyle T_{\mu \nu }}$ the energy–momentum tensor. This predicted the non-Newtonian perihelion precession of Mercury, and so had Einstein very excited. However, it was soon realized that they were inconsistent with the local conservation of energy–momentum unless the universe had a constant density of mass–energy–momentum. In other words, air, rock and even a vacuum should all have the same density. This inconsistency with observation sent Einstein back to the drawing board. However, the solution was all but obvious, and on November 25, 1915 Einstein presented the actual Einstein field equations to the Prussian Academy of Sciences:

${\displaystyle R_{\mu \nu }-{1 \over 2}Rg_{\mu \nu }=T_{\mu \nu }}$,

where ${\displaystyle R}$ is the Ricci scalar and ${\displaystyle g_{\mu \nu }}$ the metric tensor. With the publication of the field equations, the issue became one of solving them for various cases and interpreting the solutions. This and experimental verification have dominated general relativity research ever since.

Einstein and Hilbert

Although Einstein is credited with finding the field equations, the German mathematician David Hilbert published them in an article before Einstein's article. This has resulted in accusations of plagiarism against Einstein, although not from Hilbert, and assertions that the field equations should be called the "Einstein–Hilbert field equations". However, Hilbert did not press his claim for priority and some have asserted that Einstein submitted the correct equations before Hilbert amended his own work to include them. This suggests that Einstein developed the correct field equations first, though Hilbert may have reached them later independently (or even learned of them afterwards through his correspondence with Einstein). However, others have criticized those assertions.

Sir Arthur Eddington

In the early years after Einstein's theory was published, Sir Arthur Eddington lent his considerable prestige in the British scientific establishment in an effort to champion the work of this German scientist. Because the theory was so complex and abstruse (even today it is popularly considered the pinnacle of scientific thinking; in the early years it was even more so), it was rumored that only three people in the world understood it. There was an illuminating, though probably apocryphal, anecdote about this. As related by Ludwik Silberstein, during one of Eddington's lectures he asked "Professor Eddington, you must be one of three persons in the world who understands general relativity." Eddington paused, unable to answer. Silberstein continued "Don't be modest, Eddington!" Finally, Eddington replied "On the contrary, I'm trying to think who the third person is."

Solutions

The Schwarzschild solution

Since the field equations are non-linear, Einstein assumed that they were unsolvable. However, Karl Schwarzschild discovered in 1915 and published in 1916 an exact solution for the case of a spherically symmetric spacetime surrounding a massive object in spherical coordinates. This is now known as the Schwarzschild solution. Since then, many other exact solutions have been found.

The expanding universe and the cosmological constant

In 1922, Alexander Friedmann found a solution in which the universe may expand or contract, and later Georges Lemaître derived a solution for an expanding universe. However, Einstein believed that the universe was apparently static, and since a static cosmology was not supported by the general relativistic field equations, he added a cosmological constant Λ to the field equations, which became

${\displaystyle R_{\mu \nu }-{1 \over 2}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=T_{\mu \nu }}$.

This permitted the creation of steady-state solutions, but they were unstable: the slightest perturbation of a static state would result in the universe expanding or contracting. In 1929, Edwin Hubble found evidence for the idea that the universe is expanding. This resulted in Einstein dropping the cosmological constant, referring to it as "the biggest blunder in my career". At the time, it was an ad hoc hypothesis to add in the cosmological constant, as it was only intended to justify one result (a static universe).

More exact solutions

Progress in solving the field equations and understanding the solutions has been ongoing. The solution for a spherically symmetric charged object was discovered by Reissner and later rediscovered by Nordström, and is called the Reissner–Nordström solution. The black hole aspect of the Schwarzschild solution was very controversial, and Einstein did not believe that singularities could be real. However, in 1957 (two years after Einstein's death in 1955), Martin Kruskal published a proof that black holes are called for by the Schwarzschild Solution. Additionally, the solution for a rotating massive object was obtained by Kerr in the 1960s and is called the Kerr solution. The Kerr–Newman solution for a rotating, charged massive object was published a few years later.

Testing the theory

The perihelion precession of Mercury was the first evidence that general relativity is correct. Sir Arthur Stanley Eddington's 1919 expedition in which he confirmed Einstein's prediction for the deflection of light by the Sun during the total solar eclipse of 29 May 1919 helped to cement the status of general relativity as a likely true theory. Since then many observations have confirmed the correctness of general relativity. These include studies of binary pulsars, observations of radio signals passing the limb of the Sun, and even the GPS system.
Gravitational waves are ripples in the curvature of spacetime which propagate as waves, travelling outward from the source. They were first detected in September 2015 by the Advanced LIGO team from the merging of a pair of black holes.

Alternative theories

There have been various attempts to find modifications to general relativity. The most famous of these are the Brans–Dicke theory (also known as scalar-tensor theory), and Rosen's bimetric theory. Both of these theories proposed changes to the field equations of general relativity, and both suffer from these changes permitting the presence of bipolar gravitational radiation. As a result, Rosen's original theory has been refuted by observations of binary pulsars. As for Brans–Dicke (which has a tunable parameter ω such that ω = ∞ is the same as general relativity), the amount by which it can differ from general relativity has been severely constrained by these observations.

In addition, general relativity is inconsistent with quantum mechanics, the physical theory that describes the wave–particle duality of matter, and quantum mechanics does not currently describe gravitational attraction at relevant (microscopic) scales. There is a great deal of speculation in the physics community as to the modifications that might be needed to both general relativity and quantum mechanics in order to unite them consistently. The speculative theory that unites general relativity and quantum mechanics is usually called quantum gravity, prominent examples of which include String Theory and Loop Quantum Gravity.

More about GR history

Kip Thorne identifies the "golden age of general relativity" as the period roughly from 1960 to 1975 during which the study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics. During this period, many of the concepts and terms which continue to inspire the imagination of gravitation researchers and the general public were introduced, including black holes and 'gravitational singularity'. At the same time, in a closely related development, the study of physical cosmology entered the mainstream and the Big Bang became well established.

Fulvio Melia refers frequently to the "golden age of relativity" in his book Cracking the Einstein Code. Andrzej Trautman hosted a relativity conference in Warsaw in 1962 to which Melia refers:

General relativity moved very successfully from that meeting in Warsaw, hot on the heels of the Pound–Rebka experiment, and entered its golden age of discovery that lasted into the mid-1970’s.

Roy Kerr, protagonist of the book, contributed an Afterword, saying of the book: "It is a remarkable piece of writing capturing beautifully the period we now refer to as the golden age of relativity."

Particle in a box

From Wikipedia, the free encyclopedia

 

Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not.

In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a ball trapped inside a large box, the particle can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometres), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes.

The particle in a box model is one of the very few problems in quantum mechanics which can be solved analytically, without approximations. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. It serves as a simple illustration of how energy quantization (energy levels), which are found in more complicated quantum systems such as atoms and molecules, come about. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems.

One-dimensional solution

The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential.

The simplest form of the particle in a box model considers a one-dimensional system. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end. The walls of a one-dimensional box may be visualised as regions of space with an infinitely large potential energy. Conversely, the interior of the box has a constant, zero potential energy. This means that no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as

${\displaystyle V(x)={\begin{cases}0,&x_{c}-{\tfrac {L}{2}}$

where L is the length of the box, x_c is the location of the center of the box and x is the position of the particle within the box. Simple cases include the centered box (x_c = 0 ) and the shifted box (x_c = L/2 ).

Position wave function

In quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction. The wavefunction ${\displaystyle \psi (x,t)}$ can be found by solving the Schrödinger equation for the system

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t),}$

where ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle m}$ is the mass of the particle, ${\displaystyle i}$ is the imaginary unit and ${\displaystyle t}$ is time.

Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space and time with the same form as a free particle:

Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space and time with the same form as a free particle:

${\displaystyle \psi (x,t)=[A\sin(kx)+B\cos(kx)]\mathrm {e} ^{-i\omega t},}$

(1)

where ${\displaystyle A}$ and ${\displaystyle B}$ are arbitrary complex numbers. The frequency of the oscillations through space and time is given by the wavenumber ${\displaystyle k}$ and the angular frequency ${\displaystyle \omega }$ respectively. These are both related to the total energy of the particle by the expression

which is known as the dispersion relation for a free particle. Here one must notice that now, since the particle is not entirely free but under the influence of a potential (the potential V described above), the energy of the particle given above is not the same thing as ${\displaystyle {\frac {p^{2}}{2m}}}$ where p is the momentum of the particle, and thus the wavenumber k above actually describes the energy states of the particle, not the momentum states (i.e. it turns out that the momentum of the particle is not given by ${\displaystyle p=\hbar k}$). In this sense, it is quite dangerous to call the number k a wavenumber, since it is not related to momentum like "wavenumber" usually is. The rationale for calling k the wavenumber is that it enumerates the number of crests that the wavefunction has inside the box, and in this sense it is a wavenumber. This discrepancy can be seen more clearly below, when we find out that the energy spectrum of the particle is discrete (only discrete values of energy are allowed) but the momentum spectrum is continuous (momentum can vary continuously) and in particular, the relation ${\displaystyle E={\frac {p^{2}}{2m}}}$ for the energy and momentum of the particle does not hold. As said above, the reason this relation between energy and momentum does not hold is that the particle is not free, but there is a potential V in the system, and the energy of the particle is ${\displaystyle E=T+V}$, where T is the kinetic and V the potential energy.

Initial wavefunctions for the first four states in a one-dimensional particle in a box

The size (or amplitude) of the wavefunction at a given position is related to the probability of finding a particle there by ${\displaystyle P(x,t)=|\psi (x,t)|^{2}}$. The wavefunction must therefore vanish everywhere beyond the edges of the box. Also, the amplitude of the wavefunction may not "jump" abruptly from one point to the next. These two conditions are only satisfied by wavefunctions with the form

${\displaystyle \psi _{n}(x,t)={\begin{cases}A\sin \left(k_{n}\left(x-x_{c}+{\tfrac {L}{2}}\right)\right)\mathrm {e} ^{-i\omega _{n}t},&x_{c}-{\tfrac {L}{2}}$

where 

${\displaystyle k_{n}={\frac {n\pi }{L}}}$,

and

${\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}}$,

where n is a positive integer (1,2,3,4...). For a shifted box (x_c = L/2), the solution is particularly simple. The simplest solutions, ${\displaystyle k_{n}=0}$ or ${\displaystyle A=0}$ both yield the trivial wavefunction ${\displaystyle \psi (x)=0}$, which describes a particle that does not exist anywhere in the system. Negative values of ${\displaystyle n}$ are neglected, since they give wavefunctions identical to the positive ${\displaystyle n}$ solutions except for a physically unimportant sign change. Here one sees that only a discrete set of energy values and wavenumbers k are allowed for the particle. Usually in quantum mechanics it is also demanded that the derivative of the wavefunction in addition to the wavefunction itself be continuous; here this demand would lead to the only solution being the constant zero function, which is not what we desire, so we give up this demand (as this system with infinite potential can be regarded as a nonphysical abstract limiting case, we can treat it as such and "bend the rules"). Note that giving up this demand means that the wavefunction is not a differentiable function at the boundary of the box, and thus it can be said that the wavefunction does not solve the Schrödinger equation at the boundary points ${\displaystyle x=0}$ and ${\displaystyle x=L}$ (but does solve everywhere else).

Finally, the unknown constant ${\displaystyle A}$ may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. It follows that

${\displaystyle \left|A\right|={\sqrt {\frac {2}{L}}}.}$

Thus, A may be any complex number with absolute value √2/L; these different values of A yield the same physical state, so A = √2/L can be selected to simplify.

It is expected that the eigenvalues, i.e., the energy ${\displaystyle E_{n}}$ of the box should be the same regardless of its position in space, but ${\displaystyle \psi _{n}(x,t)}$ changes. Notice that ${\displaystyle x_{c}-{\tfrac {L}{2}}{L}}$ represents a phase shift in the wave function, This phase shift has no effect when solving the Schrödinger equation, and therefore does not affect the eigenvalue.

Momentum wave function

The momentum wavefunction is proportional to the Fourier transform of the position wavefunction. With ${\displaystyle k=p/\hbar }$ (note that the parameter k describing the momentum wavefunction below is not exactly the special k_n above, linked to the energy eigenvalues), the momentum wavefunction is given by

${\displaystyle \phi _{n}(p,t)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\psi _{n}(x,t)e^{-ikx}\,dx={\sqrt {\frac {L}{\pi \hbar }}}\left({\frac {n\pi }{n\pi +kL}}\right)\,{\textrm {sinc}}\left({\tfrac {1}{2}}(n\pi -kL)\right)e^{-ikx_{c}},}$

where sinc is the cardinal sine sinc function, sinc(x)=sin(x)/x. For the centered box (x_c= 0), the solution is real and particularly simple, since the phase factor on the right reduces to unity. (With care, it can be written as an even function of p.)

It can be seen that the momentum spectrum in this wave packet is continuous, and one may conclude that for the energy state described by the wavenumber k_n, the momentum can, when measured, also attain other values beyond ${\displaystyle p=\pm \hbar k_{n}}$.

Hence, it also appears that, since the energy is ${\displaystyle E_{n}={\frac {\hbar ^{2}k_{n}^{2}}{2m}}}$ for the nth eigenstate, the relation ${\displaystyle E={\frac {p^{2}}{2m}}}$ does not strictly hold for the measured momentum p; the energy eigenstate ${\displaystyle \psi _{n}}$ is not a momentum eigenstate, and, in fact, not even a superposition of two momentum eigenstates, as one might be tempted to imagine from equation (1) above: peculiarly, it has no well-defined momentum before measurement!

Position and momentum probability distributions

In classical physics, the particle can be detected anywhere in the box with equal probability. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wavefunction as ${\displaystyle P(x)=|\psi (x)|^{2}.}$ For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by

${\displaystyle P_{n}(x,t)={\begin{cases}{\frac {2}{L}}\sin ^{2}(k_{n}(x-x_{c}+{\tfrac {L}{2}})),&x_{c}-L/2$

Thus, for any value of n greater than one, there are regions within the box for which ${\displaystyle P(x)=0}$, indicating that spatial nodes exist at which the particle cannot be found.

In quantum mechanics, the average, or expectation value of the position of a particle is given by

${\displaystyle \langle x\rangle =\int _{-\infty }^{\infty }xP_{n}(x)\,\mathrm {d} x.}$

For the steady state particle in a box, it can be shown that the average position is always ${\displaystyle \langle x\rangle =x_{c}}$, regardless of the state of the particle. For a superposition of states, the expectation value of the position will change based on the cross term which is proportional to ${\displaystyle \cos(\omega t)}$.

The variance in the position is a measure of the uncertainty in position of the particle:

${\displaystyle \mathrm {Var} (x)=\int _{-\infty }^{\infty }(x-\langle x\rangle )^{2}P_{n}(x)\,dx={\frac {L^{2}}{12}}\left(1-{\frac {6}{n^{2}\pi ^{2}}}\right)}$

The probability density for finding a particle with a given momentum is derived from the wavefunction as ${\displaystyle P(x)=|\phi (x)|^{2}}$. As with position, the probability density for finding the particle at a given momentum depends upon its state, and is given by

${\displaystyle P_{n}(p)={\frac {L}{\pi \hbar }}\left({\frac {n\pi }{n\pi +kL}}\right)^{2}\,{\textrm {sinc}}^{2}\left({\tfrac {1}{2}}(n\pi -kL)\right)}$

where, again, ${\displaystyle k=p/\hbar }$. The expectation value for the momentum is then calculated to be zero, and the variance in the momentum is calculated to be:

${\displaystyle \mathrm {Var} (p)=\left({\frac {\hbar n\pi }{L}}\right)^{2}}$

The uncertainties in position and momentum (${\displaystyle \Delta x}$ and ${\displaystyle \Delta p}$) are defined as being equal to the square root of their respective variances, so that:

${\displaystyle \Delta x\Delta p={\frac {\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}}$

This product increases with increasing n, having a minimum value for n=1. The value of this product for n=1 is about equal to 0.568 ${\displaystyle \hbar }$ which obeys the Heisenberg uncertainty principle, which states that the product will be greater than or equal to ${\displaystyle \hbar /2}$.
 
Another measure of uncertainty in position is the information entropy of the probability distribution H_x:

${\displaystyle H_{x}=\int _{-\infty }^{\infty }P_{n}(x)\log(P_{n}(x)x_{0})\,dx=\log \left({\frac {2L}{e\,x_{0}}}\right)}$

where x₀ is an arbitrary reference length.

Another measure of uncertainty in momentum is the information entropy of the probability distribution H_p:

${\displaystyle H_{p}(n)=\int _{-\infty }^{\infty }P_{n}(p)\log(P_{n}(p)p_{0})\,dp}$

where p₀ is an arbitrary reference momentum. The integral is difficult to express analytically for general n, but in the limit as n approaches infinity:

${\displaystyle \lim _{n\to \infty }H_{p}(n)=\log \left({\frac {4\pi \hbar \,e^{2(1-\gamma )}}{L\,p_{0}}}\right)}$

where γ is Euler's constant. The quantum mechanical entropic uncertainty principle states that for ${\displaystyle x_{0}\,p_{0}=\hbar }$

${\displaystyle H_{x}+H_{p}(n)\geq \log(e\,\pi )\approx 2.14473...}$

For ${\displaystyle x_{0}\,p_{0}=\hbar }$, the sum of the position and momentum entropies yields:

${\displaystyle H_{x}+H_{p}(\infty )=\log(8\pi \,e^{1-2\gamma })\approx 3.06974...}$

which satisfies the quantum entropic uncertainty principle.

Energy levels

The energy of a particle in a box (black circles) and a free particle (grey line) both depend upon wavenumber in the same way. However, the particle in a box may only have certain, discrete energy levels.

The energies which correspond with each of the permitted wavenumbers may be written as

${\displaystyle E_{n}={\frac {n^{2}\hbar ^{2}\pi ^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}}$.

The energy levels increase with ${\displaystyle n^{2}}$, meaning that high energy levels are separated from each other by a greater amount than low energy levels are. The lowest possible energy for the particle (its zero-point energy) is found in state 1, which is given by

${\displaystyle E_{1}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}.}$

The particle, therefore, always has a positive energy. This contrasts with classical systems, where the particle can have zero energy by resting motionlessly. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by

${\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}}}$

It can be shown that the uncertainty in the position of the particle is proportional to the width of the box. Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box.^[9] The kinetic energy of a particle is given by ${\displaystyle E=p^{2}/(2m)}$, and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above.

Higher-dimensional boxes

(Hyper)rectangular walls

The wavefunction of a 2D well with n_x=4 and n_y=4

If a particle is trapped in a two-dimensional box, it may freely move in the ${\displaystyle x}$ and ${\displaystyle y}$-directions, between barriers separated by lengths ${\displaystyle L_{x}}$ and ${\displaystyle L_{y}}$ respectively. For a centered box, the position wave function may be written including the length of the box as ${\displaystyle \psi _{n}(x,t,L)}$. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies for a centered box are given respectively by

${\displaystyle \psi _{n_{x},n_{y}}=\psi _{n_{x}}(x,t,L_{x})\psi _{n_{y}}(y,t,L_{y})}$,

${\displaystyle E_{n_{x},n_{y}}={\frac {\hbar ^{2}k_{n_{x},n_{y}}^{2}}{2m}}}$,

where the two-dimensional wavevector is given by

${\displaystyle \mathbf {k_{n_{x},n_{y}}} =k_{n_{x}}\mathbf {\hat {x}} +k_{n_{y}}\mathbf {\hat {y}} ={\frac {n_{x}\pi }{L_{x}}}\mathbf {\hat {x}} +{\frac {n_{y}\pi }{L_{y}}}\mathbf {\hat {y}} }$.

For a three dimensional box, the solutions are

${\displaystyle \psi _{n_{x},n_{y},n_{z}}=\psi _{n_{x}}(x,t,L_{x})\psi _{n_{y}}(y,t,L_{y})\psi _{n_{z}}(z,t,L_{z})}$,

${\displaystyle E_{n_{x},n_{y},n_{z}}={\frac {\hbar ^{2}k_{n_{x},n_{y},n_{z}}^{2}}{2m}}}$,

where the three-dimensional wavevector is given by:

${\displaystyle \mathbf {k_{n_{x},n_{y},n_{z}}} =k_{n_{x}}\mathbf {\hat {x}} +k_{n_{y}}\mathbf {\hat {y}} +k_{n_{z}}\mathbf {\hat {z}} ={\frac {n_{x}\pi }{L_{x}}}\mathbf {\hat {x}} +{\frac {n_{y}\pi }{L_{y}}}\mathbf {\hat {y}} +{\frac {n_{z}\pi }{L_{z}}}\mathbf {\hat {z}} }$.

In general for an n-dimensional box, the solutions are

${\displaystyle \psi =\prod _{i}\psi _{n_{i}}(x_{i},t,L_{i})}$

The 1-dimensional momentum wave functions may likewise be represented by ${\displaystyle \phi _{n}(x,t,L_{x})}$ and the momentum wave function for an n-dimensional centered box is then:

${\displaystyle \phi =\prod _{i}\phi _{n_{i}}(k_{i},t,L_{i})}$

An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. ${\displaystyle L_{x}=L_{y}}$), there are multiple wavefunctions corresponding to the same total energy. For example, the wavefunction with ${\displaystyle n_{x}=2,n_{y}=1}$ has the same energy as the wavefunction with ${\displaystyle n_{x}=1,n_{y}=2}$. This situation is called degeneracy and for the case where exactly two degenerate wavefunctions have the same energy that energy level is said to be doubly degenerate. Degeneracy results from symmetry in the system. For the above case two of the lengths are equal so the system is symmetric with respect to a 90° rotation.

More complicated wall shapes

The wavefunction for a quantum-mechanical particle in a box whose walls have arbitrary shape is given by the Helmholtz equation subject to the boundary condition that the wavefunction vanishes at the walls. These systems are studied in the field of quantum chaos for wall shapes whose corresponding dynamical billiard tables are non-integrable.

Applications

Because of its mathematical simplicity, the particle in a box model is used to find approximate solutions for more complex physical systems in which a particle is trapped in a narrow region of low electric potential between two high potential barriers. These quantum well systems are particularly important in optoelectronics, and are used in devices such as the quantum well laser, the quantum well infrared photodetector and the quantum-confined Stark effect modulator. It is also used to model a lattice in the Kronig-Penny model and for a finite metal with the free electron approximation.

Conjugated polyenes

β-carotene is a conjugated polyene

Conjugated polyene systems can be modeled using particle in a box. The conjugated system of electrons can be modeled as a one dimensional box with length equal to the total bond distance from one terminus of the polyene to the other. In this case each pair of electrons in each π bond corresponds to one energy level. The energy difference between two energy levels, n_f and n_i is:

${\displaystyle \Delta E={\frac {(n_{f}^{2}-n_{i}^{2})h^{2}}{8mL^{2}}}}$

The difference between the ground state energy, n, and the first excited state, n+1, corresponds to the energy required to excite the system. This energy has a specific wavelength, and therefore color of light, related by:

${\displaystyle \lambda ={\frac {hc}{\Delta E}}}$

A common example of this phenomenon is in β-carotene. β-carotene (C₄₀H₅₆) is a conjugated polyene with an orange color and a molecular length of approximately 3.8 nm (though its chain length is only approximately 2.4 nm). Due to β-carotene's high level of conjugation, electrons are dispersed throughout the length of the molecule, allowing one to model it as a one-dimensional particle in a box. β-carotene has 11 carbon-carbon double bonds in conjugation; each of those double bonds contains two π-electrons, therefore β-carotene has 22 π-electrons. With two electrons per energy level, β-carotene can be treated as a particle in a box at energy level n=11. Therefore, the minimum energy needed to excite an electron to the next energy level can be calculated, n=12, as follows (recalling that the mass of an electron is 9.109 × 10⁻³¹ kg):

${\displaystyle \Delta E={\frac {(n_{f}^{2}-n_{i}^{2})h^{2}}{8mL^{2}}}}$
${\displaystyle ={\frac {(12^{2}-11^{2})h^{2}}{8mL^{2}}}}$
${\displaystyle =2.3658\times 10^{-19}{\text{ J}}}$

Using the previous relation of wavelength to energy, recalling both Planck's constant h and the speed of light c:

${\displaystyle \lambda ={\tfrac {hc}{\Delta E}}}$
${\displaystyle =0.00000084{\text{ m}}=840{\text{ nm}}}$

This indicates that β-carotene primarily absorbs light in the infrared spectrum, therefore it would appear white to a human eye. However the observed wavelength is 450 nm, indicating that the particle in a box is not a perfect model for this system.

Quantum well laser

The particle in a box model can be applied to quantum well lasers, which are laser diodes consisting of one semiconductor “well” material sandwiched between two other semiconductor layers of different material . Because the layers of this sandwich are very thin (the middle layer is typically about 100 Å thick), quantum confinement effects can be observed. The idea that quantum effects could be harnessed to create better laser diodes originated in the 1970s. The quantum well laser was patented in 1976 by R. Dingle and C. H. Henry.

Specifically, the quantum well’s behavior can be represented by the particle in a finite well model. Two boundary conditions must be selected. The first is that the wave function must be continuous. Often, the second boundary condition is chosen to be the derivative of the wave function must be continuous across the boundary, but in the case of the quantum well the masses are different on either side of the boundary. Instead, the second boundary condition is chosen to conserve particle flux as${\displaystyle (1/m)d\phi /dz}$, which is consistent with experiment. The solution to the finite well particle in a box must be solved numerically, resulting in wave functions that are sine functions inside the quantum well and exponentially decaying functions in the barriers. This quantization of the energy levels of the electrons allows a quantum well laser to emit light more efficiently than conventional semiconductor lasers.

Due to their small size, quantum dots do not showcase the bulk properties of the specified semi-conductor but rather show quantised energy states. This effect is known as the quantum confinement and has led to numerous applications of quantum dots such as the quantum well laser.

Researchers at Princeton University have recently built a quantum well laser which is no bigger than a grain of rice. The laser is powered by a single electron which passes through two quantum dots; a double quantum dot. The electron moves from a state of higher energy, to a state of lower energy whilst emitting photons in the microwave region. These photons bounce off mirrors to create a beam of light; the laser.

The quantum well laser is heavily based on the interaction between light and electrons. This relationship is a key component in quantum mechanical theories which include the De Broglie Wavelength and Particle in a box. The double quantum dot allows scientists to gain full control over the movement of an electron which consequently results in the production of a laser beam.

Quantum dots

Quantum dots are extremely small semiconductors (on the scale of nanometers). They display quantum confinement in that the electrons cannot escape the “dot”, thus allowing particle-in-a-box approximations to be applied. Their behavior can be described by three-dimensional particle-in-a-box energy quantization equations.

The energy gap of a quantum dot is the energy gap between its valence and conduction bands. This energy gap ${\displaystyle \Delta E(r)}$ is equal to the band gap of the bulk material ${\displaystyle E_{\text{gap}}}$ plus the energy equation derived from particle-in-a-box, which gives the energy for electrons and holes. This can be seen in the following equation, where ${\displaystyle m_{e}^{*}}$ and ${\displaystyle m_{h}^{*}}$ are the effective masses of the electron and hole, ${\displaystyle r}$ is radius of the dot, and ${\displaystyle h}$ is Planck's constant:

${\displaystyle \Delta E(r)=E_{\text{gap}}+\left({\frac {h^{2}}{8r^{2}}}\right)\left({\frac {1}{m_{e}^{*}}}+{\frac {1}{m_{h}^{*}}}\right)}$

Hence, the energy gap of the quantum dot is inversely proportional to the square of the “length of the box,” i.e. the radius of the quantum dot.

Manipulation of the band gap allows for the absorption and emission of specific wavelengths of light, as energy is inversely proportional to wavelength. The smaller the quantum dot, the larger the band gap and thus the shorter the wavelength absorbed.

Different semiconducting materials are used to synthesize quantum dots of different sizes and therefore emit different wavelengths of light. Materials that normally emit light in the visible region are often used and their sizes are fine-tuned so that certain colors are emitted. Typical substances used to synthesize quantum dots are cadmium (Cd) and selenium (Se). For example, when the electrons of two nanometer CdSe quantum dots relax after excitation, blue light is emitted. Similarly, red light is emitted in four nanometer CdSe quantum dots.

Quantum dots have a variety of functions including but not limited to fluorescent dyes, transistors, LEDs, solar cells, and medical imaging via optical probes.

One function of quantum dots is their use in lymph node mapping, which is feasible due to their unique ability to emit light in the near infrared (NIR) region. Lymph node mapping allows surgeons to track if and where cancerous cells exist.

Quantum dots are useful for these functions due to their emission of brighter light, excitation by a wide variety of wavelengths, and higher resistance to light than other substances.