Photon polarization

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

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well. Polarization is an example of a qubit degree of freedom, which forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in the description. The quantum polarization state vector for the photon, for instance, is identical with the Jones vector, usually used to describe the polarization of a classical wave. Unitary operators emerge from the classical requirement of the conservation of energy of a classical wave propagating through lossless media that alter the polarization state of the wave. Hermitian operators then follow for infinitesimal transformations of a classical polarization state.

Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of the experiments can be performed with polaroid sunglass lenses.

The connection with quantum mechanics is made through the identification of a minimum packet size, called a photon, for energy in the electromagnetic field. The identification is based on the theories of Planck and the interpretation of those theories by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.

Polarization of classical electromagnetic waves

Main article: Polarization (waves)

Polarization states

Linear polarization

Main article: Linear polarization

Effect of a polarizer on reflection from mud flats. In the first picture, the polarizer is rotated to minimize the effect; in the second it is rotated 90° to maximize it: almost all reflected sunlight is eliminated.

The wave is linearly polarized (or plane polarized) when the phase angles ${\displaystyle {\alpha }_x,{\alpha }_y}$ are equal, $${\displaystyle {\alpha }_x={\alpha }_y\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\alpha .}$$

This represents a wave with phase ${\displaystyle \alpha }$ polarized at an angle ${\displaystyle \theta }$ with respect to the x axis. In this case the Jones vector $${\displaystyle |\psi ⟩=\left(\begin{array}{c}\cos \theta \exp \left(i{\alpha }_x\right)\\ \sin \theta \exp \left(i{\alpha }_y\right)\end{array}\right)}$$ can be written with a single phase: $${\displaystyle |\psi ⟩=\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)\exp \left(i\alpha \right).}$$

The state vectors for linear polarization in x or y are special cases of this state vector.

If unit vectors are defined such that $${\displaystyle |x⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(\begin{array}{c}1\\ 0\end{array}\right)}$$ and $${\displaystyle |y⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(\begin{array}{c}0\\ 1\end{array}\right)}$$ then the linearly polarized polarization state can be written in the "x–y basis" as $${\displaystyle |\psi ⟩=\cos \theta \exp \left(i\alpha \right)|x⟩+\sin \theta \exp \left(i\alpha \right)|y⟩={\psi }_x|x⟩+{\psi }_y|y⟩.}$$

Circular polarization

Main article: Circular polarization

If the phase angles ${\displaystyle {\alpha }_x}$ and ${\displaystyle {\alpha }_y}$ differ by exactly ${\displaystyle \pi /2}$ and the x amplitude equals the y amplitude the wave is circularly polarized. The Jones vector then becomes $${\displaystyle |\psi ⟩=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ \pm i\end{array}\right)\exp \left(i{\alpha }_x\right)}$$ where the plus sign indicates left circular polarization and the minus sign indicates right circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x–y plane.

If unit vectors are defined such that $${\displaystyle |\mathrm{R}⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)}$$ and $${\displaystyle |\mathrm{L}⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -i\end{array}\right)}$$ then an arbitrary polarization state can be written in the "R–L basis" as $${\displaystyle |\psi ⟩={\psi }_{\mathrm{R}}|\mathrm{R}⟩+{\psi }_{\mathrm{L}}|\mathrm{L}⟩}$$ where $${\displaystyle {\psi }_{\mathrm{R}}=⟨\mathrm{R}|\psi ⟩=\frac{1}{\sqrt{2}}\left(\cos \theta \exp (i{\alpha }_x)-i\sin \theta \exp (i{\alpha }_y)\right)}$$ and $${\displaystyle {\psi }_{\mathrm{L}}=⟨\mathrm{L}|\psi ⟩=\frac{1}{\sqrt{2}}\left(\cos \theta \exp (i{\alpha }_x)+i\sin \theta \exp (i{\alpha }_y)\right).}$$

We can see that $${\displaystyle 1=|{\psi }_{\mathrm{R}}{|}^2+|{\psi }_{\mathrm{L}}{|}^2.}$$

Elliptical polarization

Main article: Elliptical polarization

The general case in which the electric field rotates in the x–y plane and has variable magnitude is called elliptical polarization. The state vector is given by $${\displaystyle |\psi ⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(\begin{array}{c}{\psi }_x\\ {\psi }_y\end{array}\right)=\left(\begin{array}{c}\cos \theta \exp \left(i{\alpha }_x\right)\\ \sin \theta \exp \left(i{\alpha }_y\right)\end{array}\right).}$$

Geometric visualization of an arbitrary polarization state

To get an understanding of what a polarization state looks like, one can observe the orbit that is made if the polarization state is multiplied by a phase factor of ${\displaystyle e^{i\omega t}}$ and then having the real parts of its components interpreted as x and y coordinates respectively. That is: $${\displaystyle \left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)=\left(\begin{array}{c}\mathrm{\Re }(e^{i\omega t}{\psi }_x)\\ \mathrm{\Re }(e^{i\omega t}{\psi }_y)\end{array}\right)=\mathrm{\Re }\left[e^{i\omega t}\left(\begin{array}{c}{\psi }_x\\ {\psi }_y\end{array}\right)\right]=\mathrm{\Re }\left(e^{i\omega t}|\psi ⟩\right).}$$

If only the traced out shape and the direction of the rotation of (x(t), y(t)) is considered when interpreting the polarization state, i.e. only $${\displaystyle M(|\psi ⟩)=\left\{(x(t),y(t))\right|\mathrm{\forall }t\}}$$ (where x(t) and y(t) are defined as above) and whether it is overall more right circularly or left circularly polarized (i.e. whether |ψ_R| > |ψ_L| or vice versa), it can be seen that the physical interpretation will be the same even if the state is multiplied by an arbitrary phase factor, since $${\displaystyle M(e^{i\alpha }|\psi ⟩)=M(|\psi ⟩),\alpha \in \mathbb{R}}$$ and the direction of rotation will remain the same. In other words, there is no physical difference between two polarization states ${\displaystyle |\psi ⟩}$ and ${\displaystyle e^{i\alpha }|\psi ⟩}$, between which only a phase factor differs.

It can be seen that for a linearly polarized state, M will be a line in the xy plane, with length 2 and its middle in the origin, and whose slope equals to tan(θ). For a circularly polarized state, M will be a circle with radius 1/√2 and with the middle in the origin.

Energy, momentum, and angular momentum of a classical electromagnetic wave

Energy density of classical electromagnetic waves

Energy in a plane wave

Main article: Energy density

The energy per unit volume in classical electromagnetic fields is (cgs units) and also Planck units: $${\displaystyle {\mathcal{E}}_c=\frac{1}{8\pi }\left[{\mathbf{E}}^2(\mathbf{r},t)+{\mathbf{B}}^2(\mathbf{r},t)\right].}$$

For a plane wave, this becomes: $${\displaystyle {\mathcal{E}}_c=\frac{\mid \mathbf{E}{\mid }^2}{8\pi }}$$ where the energy has been averaged over a wavelength of the wave.

Fraction of energy in each component

The fraction of energy in the x component of the plane wave is $${\displaystyle f_x=\frac{|\mathbf{E}{|}^2{\cos }^2\theta }{|\mathbf{E}{|}^2}={\psi }_x^{\ast }{\psi }_x={\cos }^2\theta }$$ with a similar expression for the y component resulting in ${\displaystyle f_y={\sin }^2\theta }$.

The fraction in both components is $${\displaystyle {\psi }_x^{\ast }{\psi }_x+{\psi }_y^{\ast }{\psi }_y=⟨\psi |\psi ⟩=1.}$$

Momentum density of classical electromagnetic waves

The momentum density is given by the Poynting vector $${\displaystyle \mathcal{P}=\frac{1}{4\pi c}\mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t).}$$

For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density: $${\displaystyle {\mathcal{P}}_{z}c={\mathcal{E}}_c.}$$

The momentum density has been averaged over a wavelength.

Angular momentum density of classical electromagnetic waves

Main article: Angular momentum of light

Electromagnetic waves can have both orbital and spin angular momentum.^[1] The total angular momentum density is $${\displaystyle \mathcal{L}=\mathbf{r}\times \mathcal{P}=\frac{1}{4\pi c}\mathbf{r}\times \left[\mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t)\right].}$$

For a sinusoidal plane wave propagating along ${\displaystyle z}$ axis the orbital angular momentum density vanishes. The spin angular momentum density is in the ${\displaystyle z}$ direction and is given by $${\displaystyle \mathcal{L}=\frac{|\mathbf{E}{|}^2}{8\pi \omega }\left({\left|⟨\mathrm{R}|\psi ⟩\right|}^2-{\left|⟨\mathrm{L}|\psi ⟩\right|}^2\right)=\frac{1}{\omega }{\mathcal{E}}_c\left(|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2\right)}$$ where again the density is averaged over a wavelength.

Optical filters and crystals

Passage of a classical wave through a polaroid filter

Linear polarization

A linear filter transmits one component of a plane wave and absorbs the perpendicular component. In that case, if the filter is polarized in the x direction, the fraction of energy passing through the filter is $${\displaystyle f_x={\psi }_x^{\ast }{\psi }_x={\cos }^2\theta .}$$

Example of energy conservation: Passage of a classical wave through a birefringent crystal

An ideal birefringent crystal transforms the polarization state of an electromagnetic wave without loss of wave energy. Birefringent crystals therefore provide an ideal test bed for examining the conservative transformation of polarization states. Even though this treatment is still purely classical, standard quantum tools such as unitary and Hermitian operators that evolve the state in time naturally emerge.

Initial and final states

A birefringent crystal is a material that has an optic axis with the property that the light has a different index of refraction for light polarized parallel to the axis than it has for light polarized perpendicular to the axis. Light polarized parallel to the axis are called "extraordinary rays" or "extraordinary photons", while light polarized perpendicular to the axis are called "ordinary rays" or "ordinary photons". If a linearly polarized wave impinges on the crystal, the extraordinary component of the wave will emerge from the crystal with a different phase than the ordinary component. In mathematical language, if the incident wave is linearly polarized at an angle ${\displaystyle theta}$ with respect to the optic axis, the incident state vector can be written $${\displaystyle |\psi ⟩=\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)}$$ and the state vector for the emerging wave can be written $${\displaystyle |{\psi }^{\prime }⟩=\left(\begin{array}{c}\cos \theta \exp \left(i{\alpha }_x\right)\\ \sin \theta \exp \left(i{\alpha }_y\right)\end{array}\right)=\left(\begin{array}{cc}\exp \left(i{\alpha }_x\right)& 0\\ 0& \exp \left(i{\alpha }_y\right)\end{array}\right)\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\hat{U}|\psi ⟩.}$$

While the initial state was linearly polarized, the final state is elliptically polarized. The birefringent crystal alters the character of the polarization.

Dual of the final state

A calcite crystal laid upon a paper with some letters showing the double refraction

The initial polarization state is transformed into the final state with the operator U. The dual of the final state is given by $${\displaystyle ⟨{\psi }^{\prime }|=⟨\psi |{\hat{U}}^{†}}$$ where ${\displaystyle U^{†}}$ is the adjoint of U, the complex conjugate transpose of the matrix.

Unitary operators and energy conservation

The fraction of energy that emerges from the crystal is $${\displaystyle ⟨{\psi }^{\prime }|{\psi }^{\prime }⟩=⟨\psi |{\hat{U}}^{†}\hat{U}|\psi ⟩=⟨\psi |\psi ⟩=1.}$$

In this ideal case, all the energy impinging on the crystal emerges from the crystal. An operator U with the property that $${\displaystyle {\hat{U}}^{†}\hat{U}=I,}$$ where I is the identity operator and U is called a unitary operator. The unitary property is necessary to ensure energy conservation in state transformations.

Hermitian operators and energy conservation

Doubly refracting Calcite from Iceberg claim, Dixon, New Mexico. This 35 pound (16 kg) crystal, on display at the National Museum of Natural History, is one of the largest single crystals in the United States.

If the crystal is very thin, the final state will be only slightly different from the initial state. The unitary operator will be close to the identity operator. We can define the operator H by $${\displaystyle \hat{U}\approx I+i\hat{H}}$$ and the adjoint by $${\displaystyle {\hat{U}}^{†}\approx I-i{\hat{H}}^{†}.}$$

Energy conservation then requires $${\displaystyle I={\hat{U}}^{†}\hat{U}\approx \left(I-i{\hat{H}}^{†}\right)\left(I+i\hat{H}\right)\approx I-i{\hat{H}}^{†}+i\hat{H}.}$$

This requires that $${\displaystyle \hat{H}={\hat{H}}^{†}.}$$

Operators like this that are equal to their adjoints are called Hermitian or self-adjoint.

The infinitesimal transition of the polarization state is $${\displaystyle |{\psi }^{\prime }⟩-|\psi ⟩=i\hat{H}|\psi ⟩.}$$

Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator.

Photons: connection to quantum mechanics

Main article: Photon

Energy, momentum, and angular momentum of photons

Energy

The treatment to this point has been classical. It is a testament, however, to the generality of Maxwell's equations for electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities. The reinterpretation is based on the theories of Max Planck and the interpretation by Albert Einstein of those theories and of other experiments.^{[citation needed]}

Einstein's conclusion from early experiments on the photoelectric effect is that electromagnetic radiation is composed of irreducible packets of energy, known as photons. The energy of each packet is related to the angular frequency of the wave by the relation$${\displaystyle ϵ=\hslash \omega }$$where ${\displaystyle \hslash }$ is an experimentally determined quantity known as the reduced Planck constant. If there are ${\displaystyle N}$ photons in a box of volume ${\displaystyle V}$, the energy in the electromagnetic field is$${\displaystyle N\hslash \omega }$$and the energy density is$${\displaystyle \frac{N\hslash \omega }{V}}$$

The photon energy can be related to classical fields through the correspondence principle that states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large ${\displaystyle N}$, the quantum energy density must be the same as the classical energy density$${\displaystyle \frac{N\hslash \omega }{V}={\mathcal{E}}_c=\frac{|\mathbf{E}{|}^2}{8\pi }.}$$

The number of photons in the box is then$${\displaystyle N=\frac{V}{8\pi \hslash \omega }|\mathbf{E}{|}^2.}$$

Momentum

The correspondence principle also determines the momentum and angular momentum of the photon. For momentum$${\displaystyle {\mathcal{P}}_z=\frac{N\hslash \omega }{cV}=\frac{N\hslash k_z}{V}}$$where ${\displaystyle k_z}$ is the wave number. This implies that the momentum of a photon is$${\displaystyle p_z=\hslash k_z.}$$

Angular momentum and spin

Similarly for the spin angular momentum$${\displaystyle \mathcal{L}=\frac{1}{\omega }{\mathcal{E}}_c\left(|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2\right)=\frac{N\hslash }{V}\left(|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2\right)}$$where ${\displaystyle {\mathcal{E}}_c}$ is field strength. This implies that the spin angular momentum of the photon is$${\displaystyle l_z=\hslash \left(|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2\right).}$$the quantum interpretation of this expression is that the photon has a probability of ${\displaystyle \mid {\psi }_{\mathrm{R}}{\mid }^2}$ of having a spin angular momentum of ${\displaystyle \hslash }$ and a probability of ${\displaystyle \mid {\psi }_{\mathrm{L}}{\mid }^2}$ of having a spin angular momentum of ${\displaystyle -\hslash }$. We can therefore think of the spin angular momentum of the photon being quantized as well as the energy. The angular momentum of classical light has been verified. A photon that is linearly polarized (plane polarized) is in a superposition of equal amounts of the left-handed and right-handed states. Upon absorption by an electronic state, the angular momentum is "measured" and this superposition collapses into either right-hand or left-hand, corresponding to a raising or lowering of the angular momentum of the absorbing electronic state, respectively.

Spin operator

Main article: Spin angular momentum of light

The spin of the photon is defined as the coefficient of ${\displaystyle \hslash }$ in the spin angular momentum calculation. A photon has spin 1 if it is in the ${\displaystyle |R⟩}$ state and −1 if it is in the ${\displaystyle |L⟩}$ state. The spin operator is defined as the outer product$${\displaystyle \hat{S}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}|\mathrm{R}⟩⟨\mathrm{R}|-|\mathrm{L}⟩⟨\mathrm{L}|=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right).}$$

The eigenvectors of the spin operator are ${\displaystyle |\mathrm{R}⟩}$ and ${\displaystyle |\mathrm{L}⟩}$ with eigenvalues 1 and −1, respectively. These values are based on the point of view of the source as the convention to define circular polarization handedness.

The expected value of a spin measurement on a photon is then$${\displaystyle ⟨\psi |\hat{S}|\psi ⟩=|{\psi }_{\mathrm{R}}{|}^2-|{\psi }_{\mathrm{L}}{|}^2.}$$

An operator S has been associated with an observable quantity, the spin angular momentum. The eigenvalues of the operator are the allowed observable values. This has been demonstrated for spin angular momentum, but it is in general true for any observable quantity.

Spin states

See also: Circular polarization § Quantum mechanics

We can write the circularly polarized states as$${\displaystyle |s⟩}$$where s = 1 for ${\displaystyle |\mathrm{R}⟩}$ and s = −1 for ${\displaystyle |\mathrm{L}⟩}$. An arbitrary state can be written$${\displaystyle |\psi ⟩=\sum _{s=-1,1}{a}_s\exp \left(i{\alpha }_x-is\theta \right)|s⟩}$$where ${\displaystyle {\alpha }_1}$ and ${\displaystyle {\alpha }_{-1}}$ are phase angles, θ is the angle by which the frame of reference is rotated, and$${\displaystyle \sum _{s=-1,1}|a_s{|}^2=1.}$$

Spin and angular momentum operators in differential form

When the state is written in spin notation, the spin operator can be written$${\displaystyle {\hat{S}}_d\to i\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}$$$${\displaystyle {\hat{S}}_d^{†}\to -i\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }.}$$

The eigenvectors of the differential spin operator are$${\displaystyle \exp \left(i{\alpha }_x-is\theta \right)|s⟩.}$$

To see this, note$${\displaystyle {\hat{S}}_d\exp \left(i{\alpha }_x-is\theta \right)|s⟩\to i\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\exp \left(i{\alpha }_x-is\theta \right)|s⟩=s\left[\exp \left(i{\alpha }_x-is\theta \right)|s⟩\right].}$$

The spin angular momentum operator is$${\displaystyle {\hat{l}}_z=\hslash {\hat{S}}_d.}$$

Nature of probability in quantum mechanics

Probability for a single photon

There are two ways in which probability can be applied to the behavior of photons; probability can be used to calculate the probable number of photons in a particular state, or probability can be used to calculate the likelihood of a single photon to be in a particular state. The former interpretation violates energy conservation. The latter interpretation is the viable, if nonintuitive, option. Dirac explains this in the context of the double-slit experiment:

Some time before the discovery of quantum mechanics people realized that the connection between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place. The importance of the distinction can be made clear in the following way. Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity. On the assumption that the beam is connected with the probable number of photons in it, we should have half the total number going into each component. If the two components are now made to interfere, we should require a photon in one component to be able to interfere with one in the other. Sometimes these two photons would have to annihilate one another and other times they would have to produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with probabilities for one photon gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs.
—Paul Dirac, The Principles of Quantum Mechanics, 1930, Chapter 1

Probability amplitudes

The probability for a photon to be in a particular polarization state depends on the fields as calculated by the classical Maxwell's equations. The polarization state of the photon is proportional to the field. The probability itself is quadratic in the fields and consequently is also quadratic in the quantum state of polarization. In quantum mechanics, therefore, the state or probability amplitude contains the basic probability information. In general, the rules for combining probability amplitudes look very much like the classical rules for composition of probabilities: [The following quote is from Baym, Chapter 1]

1. The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. For example, the amplitude for the x polarized photon to be right circularly polarized and for the right circularly polarized photon to pass through the y-polaroid is ${\displaystyle ⟨R|x⟩⟨y|R⟩,}$ the product of the individual amplitudes.
2. The amplitude for a process that can take place in one of several indistinguishable ways is the sum of amplitudes for each of the individual ways. For example, the total amplitude for the x polarized photon to pass through the y-polaroid is the sum of the amplitudes for it to pass as a right circularly polarized photon, ${\displaystyle ⟨y|R⟩⟨R|x⟩,}$ plus the amplitude for it to pass as a left circularly polarized photon, ${\displaystyle ⟨y|L⟩⟨L|x⟩\dots }$
3. The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.

Uncertainty principle

Main article: Uncertainty principle

Cauchy–Schwarz inequality in Euclidean space. ${\displaystyle \mathbf{V}\cdot \mathbf{W}=\Vert \mathbf{V}\Vert \Vert \mathbf{W}\Vert \cos a.}$ This implies ${\displaystyle \mathbf{V}\cdot \mathbf{W}\le \Vert \mathbf{V}\Vert \Vert \mathbf{W}\Vert .}$

Mathematical preparation

For any legal^{[clarification needed]} operators the following inequality, a consequence of the Cauchy–Schwarz inequality, is true. $${\displaystyle \frac{1}{4}{\left|⟨(\hat{A}\hat{B}-\hat{B}\hat{A})x|x⟩\right|}^2\le {\Vert \hat{A}x\Vert }^2{\Vert \hat{B}x\Vert }^2.}$$

If B A ψ and A B ψ are defined, then by subtracting the means and re-inserting in the above formula, we deduce $${\displaystyle {\mathrm{\Delta }}_{\psi }\hat{A}{\mathrm{\Delta }}_{\psi }\hat{B}\ge \frac{1}{2}\left|{⟨\left[\hat{A},\hat{B}\right]⟩}_{\psi }\right|}$$ where $${\displaystyle {⟨\hat{X}⟩}_{\psi }=⟨\psi |\hat{X}|\psi ⟩}$$ is the operator mean of observable X in the system state ψ and $${\displaystyle {\mathrm{\Delta }}_{\psi }\hat{X}=\sqrt{⟨{\hat{X}}^2{⟩}_{\psi }-⟨\hat{X}{⟩}_{\psi }^2}.}$$

Here $${\displaystyle \left[\hat{A},\hat{B}\right]\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\hat{A}\hat{B}-\hat{B}\hat{A}}$$ is called the commutator of A and B.

This is a purely mathematical result. No reference has been made to any physical quantity or principle. It simply states that the uncertainty of one operator times the uncertainty of another operator has a lower bound.

Application to angular momentum

The connection to physics can be made if we identify the operators with physical operators such as the angular momentum and the polarization angle. We have then $${\displaystyle {\mathrm{\Delta }}_{\psi }{\hat{L}}_z{\mathrm{\Delta }}_{\psi }\theta \ge \frac{\hslash }{2},}$$ which means that angular momentum and the polarization angle cannot be measured simultaneously with infinite accuracy. (The polarization angle can be measured by checking whether the photon can pass through a polarizing filter oriented at a particular angle, or a polarizing beam splitter. This results in a yes/no answer that, if the photon was plane-polarized at some other angle, depends on the difference between the two angles.)

States, probability amplitudes, unitary and Hermitian operators, and eigenvectors

Much of the mathematical apparatus of quantum mechanics appears in the classical description of a polarized sinusoidal electromagnetic wave. The Jones vector for a classical wave, for instance, is identical with the quantum polarization state vector for a photon. The right and left circular components of the Jones vector can be interpreted as probability amplitudes of spin states of the photon. Energy conservation requires that the states be transformed with a unitary operation. This implies that infinitesimal transformations are transformed with a Hermitian operator. These conclusions are a natural consequence of the structure of Maxwell's equations for classical waves.

Quantum mechanics enters the picture when observed quantities are measured and found to be discrete rather than continuous. The allowed observable values are determined by the eigenvalues of the operators associated with the observable. In the case angular momentum, for instance, the allowed observable values are the eigenvalues of the spin operator.

These concepts have emerged naturally from Maxwell's equations and Planck's and Einstein's theories. They have been found to be true for many other physical systems. In fact, the typical program is to assume the concepts of this section and then to infer the unknown dynamics of a physical system. This was done, for instance, with the dynamics of electrons. In that case, working back from the principles in this section, the quantum dynamics of particles were inferred, leading to the Schrödinger equation, a departure from Newtonian mechanics. The solution of this equation for atoms led to the explanation of the Balmer series for atomic spectra and consequently formed a basis for all of atomic physics and chemistry.

This is not the only occasion in which Maxwell's equations have forced a restructuring of Newtonian mechanics. Maxwell's equations are relativistically consistent. Special relativity resulted from attempts to make classical mechanics consistent with Maxwell's equations (see, for example, Moving magnet and conductor problem).

Psychological egoism

From Wikipedia, the free encyclopedia

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

Psychological egoism is the view that humans are always motivated by self-interest and selfishness, even in what seem to be acts of altruism. It claims that, when people choose to help others, they do so ultimately because of the personal benefits that they expect to obtain, directly or indirectly, from doing so.

This is a descriptive rather than normative view, since it only makes claims about how things are, not how they "ought to be" according to some. It is, however, related to several other normative forms of egoism, such as ethical egoism and rational egoism.

Subtypes of psychological egoism

Psychological hedonism

Main article: Psychological hedonism

A specific form of psychological egoism is psychological hedonism, the view that the ultimate motive for all voluntary human action is the desire to experience pleasure or to avoid pain.

Immediate gratification can be sacrificed for a chance of greater, future pleasure. Further, humans are not motivated strictly to avoid pain and pursue pleasure, but rather humans will endure pain to achieve the greatest net pleasure. Accordingly, all actions are tools for increasing pleasure or decreasing pain, even those defined as altruistic and those that do not cause an immediate change in satisfaction levels.

Elliott Sober argues that psychological egoist, when pressed, often has to resort to hedonism in order to maintain their position, since the supposed pleasure of acting morally can often be the only viable explanation for an altruistic action.

The most famous psychological egoists are Sextus Empiricus, Pierre Bayle, and Bernard Mandeville.

Final cause

Some theorists explain behavior motivated by self-interest without using pleasure and pain as the final causes of behavior.

Foundations

Beginning with ancient philosophy, Epicureanism claims humans live to maximize pleasure. Epicurus argued the theory of human behavior being motivated by pleasure alone is evidenced from infancy to adulthood. Humanity performs altruistic, honorable, and virtuous acts not for the sake of another or because of a moral code but rather to increase the well-being of the self.

In modern philosophy, Jeremy Bentham asserted, like Epicurus, that human behavior is governed by a need to increase pleasure and decrease pain. Bentham explicitly described what types and qualities of pain and pleasure exist, and how human motives are singularly explained using psychological hedonism. Bentham attempted to quantify psychological hedonism. Bentham endeavored to find the ideal human behavior based on hedonic calculus or the measurement of relative gains and losses in pain and pleasure to determine the most pleasurable action a human could choose in a situation.

From an evolutionary perspective, Herbert Spencer, a psychological egoist, argued that all animals primarily seek to survive and protect their lineage. Essentially, the need for the individual and for the individual's immediate family to live supersedes the others' need to live. All species attempt to maximize their own chances of survival and, therefore, well-being. Spencer asserted the best adapted creatures will have their pleasure levels outweigh their pain levels in their environments. Thus, pleasure meant an animal was fulfilling its egoist goal of self survival, and pleasure would always be pursued because species constantly strive for survival.

Contributions to modern psychology

Psychoanalysis

Whether or not Sigmund Freud was a psychological egoist, his concept of the pleasure principle borrowed much from psychological egoism and psychological hedonism in particular. The pleasure principle rules the behavior of the Id which is an unconscious force driving humans to release tension from unfulfilled desires. When Freud introduced Thanatos and its opposing force, Eros, the pleasure principle emanating from psychological hedonism became aligned with the Eros, which drives a person to satiate sexual and reproductive desires. Alternatively, Thanatos seeks the cessation of pain through death and the end of the pursuit of pleasure: thus, hedonism rules Thanatos, but it centers on the complete avoidance of pain rather than psychological hedonist function which pursues pleasure and avoids pain. Therefore, Freud believed in qualitatively different hedonisms where the total avoidance of pain hedonism and the achievement of the greatest net pleasure hedonism are separate and associated with distinct functions and drives of the human psyche. Although Eros and Thanatos are ruled by qualitatively different types of hedonism, Eros remains under the rule of Jeremy Bentham's quantitative psychological hedonism because Eros seeks the greatest net pleasure.

Behaviorism

Traditional behaviorism dictates all human behavior is explained by classical conditioning and operant conditioning. Operant conditioning works through reinforcement and punishment which adds or removes pleasure and pain to manipulate behavior. Using pleasure and pain to control behavior means behaviorists assumed the principles of psychological hedonism could be applied to predicting human behavior. For example, Thorndike's law of effect states that behaviors associated with pleasantness will be learned and those associated with pain will be extinguished. Often, behaviorist experiments using humans and animals are built around the assumption that subjects will pursue pleasure and avoid pain. Although psychological hedonism is incorporated into the fundamental principles and experimental designs of behaviorism, behaviorism itself explains and interprets only observable behavior and therefore does not theorize about the ultimate cause of human behavior. Thus, behaviorism uses but does not strictly support psychological hedonism over other understandings of the ultimate drive of human behavior.

Debate

Psychological egoism is controversial. Proponents cite evidence from introspection: reflection on one's own actions may reveal their motives and intended results to be based on self-interest. Psychological hedonists have found through numerous observations of natural human behavior that behavior can be manipulated through reward and punishment, both of which have direct effects of pain and pleasure. Also, the work of some social scientists has empirically supported this theory. Further, they claim psychological egoism posits a theory that is a more parsimonious explanation than competing theories.

Opponents have argued that psychological egoism is not more parsimonious than other theories. For example, a theory that claims altruism occurs for the sake of altruism explains altruism with less complexity than the egoistic approach. The psychological egoist asserts humans act altruistically for selfish reasons even when cost of the altruistic action is far outweighed by the reward of acting selfishly because altruism is performed to fulfill the desire of a person to act altruistically. Other critics argue that it is false either because it is an over-simplified interpretation of behavior or that there exists empirical evidence of altruistic behaviour. Recently, some have argued that evolutionary theory provides evidence against it.

Critics have stated that proponents of psychological egoism often confuse the satisfaction of their own desires with the satisfaction of their own self-regarding desires. Even though it is true that every human being seeks their own satisfaction, this sometimes may only be achieved via the well-being of their neighbor. An example of this situation could be phoning for an ambulance when a car accident has happened. In this case, the caller desires the well-being of the victim, even though the desire itself is the caller's own.

To counter this critique, psychological egoism asserts that all such desires for the well-being of others are ultimately derived from self-interest. For example, German philosopher Friedrich Nietzsche was a psychological egoist for some of his career, though he is said to have repudiated that later in his campaign against morality. He argues in §133 of The Dawn that in such cases compassionate impulses arise out of the projection of our identity unto the object of our feeling. He gives some hypothetical examples as illustrations to his thesis: that of a person, feeling horrified after witnessing a personal feud, coughing blood, or that of the impulse felt to save a person who is drowning in the water. In such cases, according to Nietzsche, there comes into play unconscious fears regarding our own safety. The suffering of another person is felt as a threat to our own happiness and sense of safety, because it reveals our own vulnerability to misfortunes, and thus, by relieving it, one could also ameliorate those personal sentiments. Essentially, proponents argue that altruism is rooted in self-interest whereas opponents claim altruism occurs for altruism's sake or is caused by a non-selfish reason.

Max Stirner

Stencil drawing of Max Stirner

Philosopher Max Stirner was an advocate for people striving towards ownness, however he rejected the concept of psychological egoism because he believed most people are slaves to a 'spook'- a framework for moral behaviour that can delude our self-interest. Examples of spooks include society and natural rights.

Stirner also uses the example of Juliet from Romeo and Juliet to counter psychological egoism. Juliet kills herself as a sacrifice for others' betterment. She is in love, and knows that by doing this she will leave her self-will unsatisfied, nevertheless she subjects herself to a higher power and prohibits herself from having what she truly wants. This demonstrates how it is possible for a person to act without satisfying one's self-interest.

Problem of apparent altruism

David Hume once wrote, "What interest can a fond mother have in view, who loses her health by assiduous attendance on her sick child, and afterwards languishes and dies of grief, when freed, by its death [the child's], from the slavery of that attendance?". It seems incorrect to describe such a mother's goal as self-interested.

Psychological egoists, however, respond that helping others in such ways is ultimately motivated by some form of self-interest, such as non-sensory satisfaction, the expectation of reciprocation, the desire to gain respect or reputation, or by the expectation of a reward in a putative afterlife. The helpful action is merely instrumental to these ultimately selfish goals.

In the ninth century, Mohammed Ibn Al-Jahm Al-Barmaki (محمد بن الجـَهْم البَرمَكي) has been quoted saying:

"No one deserves thanks from another about something he has done for him or goodness he has done, he is either willing to get a reward from God, therefore he wanted to serve himself, or he wanted to get a reward from people, therefore, he has done that to get profit for himself, or to be mentioned and praised by people, therefore, to it is also for himself, or due to his mercy and tenderheartedness, so he has simply done that goodness to pacify these feelings and treat himself."

This sort of explanation appears to be close to the view of La Rochefoucauld (and perhaps Hobbes).

According to psychological hedonism, the ultimate egoistic motive is to gain good feelings of pleasure and avoid bad feelings of pain. Other, less restricted forms of psychological egoism may allow the ultimate goal of a person to include such things as avoiding punishments from oneself or others (such as guilt or shame) and attaining rewards (such as pride, self-worth, power or reciprocal beneficial action).

Some psychologists explain empathy in terms of psychological hedonism. According to the "merge with others hypothesis", empathy increases the more an individual feels like they are one with another person, and decreases accordingly. Therefore, altruistic actions emanating from empathy, and empathy itself, are caused by making others' interests our own, and the satisfaction of their desires becomes our own, not just theirs. Both cognitive studies and neuropsychological experiments have provided evidence for this theory: as humans increase our oneness with others, our empathy increases, and as empathy increases, so too does our inclination to act altruistically. Neuropsychological studies have linked mirror neurons to humans experiencing empathy. Mirror neurons are activated both when a human (or animal) performs an action and when they observe another human (or animal) perform the same action. Researchers have found that the more these mirror neurons fire the more human subjects report empathy. From a neurological perspective, scientists argue that when a human empathizes with another, the brain operates as if the human is actually participating in the actions of the other person. Thus, when performing altruistic actions motivated by empathy, humans experience someone else's pleasure of being helped. Therefore, in performing acts of altruism, people act in their own self-interest even at a neurological level.

Criticism

Circularity

Psychological egoism has been accused of being circular: "If a person willingly performs an act, that means he derives personal enjoyment from it; therefore, people only perform acts that give them personal enjoyment." In particular, seemingly altruistic acts must be performed because people derive enjoyment from them and are therefore, in reality, egoistic. This statement is circular because its conclusion is identical to its hypothesis: it assumes that people only perform acts that give them personal enjoyment, and concludes that people only perform acts that give them personal enjoyment. This objection was tendered by William Hazlitt and Thomas Macaulay in the 19th century, and has been restated many times since. An earlier version of the same objection was made by Joseph Butler in the Fifteen Sermons.

Joel Feinberg, in his 1958 paper "Psychological Egoism", embraces a similar critique by drawing attention to the infinite regress of psychological egoism. He expounds it in the following cross-examination:

"All men desire only satisfaction."

"Satisfaction of what?"

"Satisfaction of their desires."

"Their desires for what?"

"Their desires for satisfaction."

"Satisfaction of what?"

"Their desires."

"For what?"

"For satisfaction"—etc., ad infinitum.

Evolutionary argument

In their 1998 book, Unto Others, Sober and Wilson detailed an evolutionary argument based on the likelihood for egoism to evolve under the pressures of natural selection. Specifically, they focus on the human behavior of parental care. To set up their argument, they propose two potential psychological mechanisms for this. The hedonistic mechanism is based on a parent's ultimate desire for pleasure or the avoidance of pain and a belief that caring for its offspring will be instrumental to that. The altruistic mechanism is based on an altruistic ultimate desire to care for its offspring.

Sober and Wilson argue that when evaluating the likelihood of a given trait to evolve, three factors must be considered: availability, reliability and energetic efficiency. The genes for a given trait must first be available in the gene pool for selection. The trait must then reliably produce an increase in fitness for the organism. The trait must also operate with energetic efficiency to not limit the fitness of the organism. Sober and Wilson argue that there is neither reason to suppose that an altruistic mechanism should be any less available than a hedonistic one nor reason to suppose that the content of thoughts and desires (hedonistic vs. altruistic) should impact energetic efficiency. As availability and energetic efficiency are taken to be equivalent for both mechanisms it follows that the more reliable mechanism will then be the more likely mechanism.

For the hedonistic mechanism to produce the behavior of caring for offspring, the parent must believe that the caring behavior will produce pleasure or avoidance of pain for the parent. Sober and Wilson argue that the belief also must be true and constantly reinforced, or it would not be likely enough to persist. If the belief fails then the behavior is not produced. The altruistic mechanism does not rely on belief; therefore, they argue that it would be less likely to fail than the alternative, i.e. more reliable.

Equivocation

In philosopher Derek Parfit's 2011 book On What Matters, Volume 1, Parfit presents an argument against psychological egoism that centers around an apparent equivocation between different senses of the word "want":

The word desire often refers to our sensual desires or appetites, or to our being attracted to something, by finding the thought of it appealing. I shall use ‘desire’ in a wider sense, which refers to any state of being motivated, or of wanting something to happen and being to some degree disposed to make it happen, if we can. The word want already has both these senses.

According to Parfit, the argument for psychological egoism fails, because it uses the word want first in the wide sense and then in the narrow sense. If I voluntarily gave up my life to save the lives of several strangers, my act would not be selfish, though I would be doing what in the wide sense I wanted to do.

Barriers to pro-environmental behaviour

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Barriers_to_pro-environmental_behaviour

Pro-environmental behaviour is behaviour that people consciously choose in order to minimize the negative impact of their actions on the environment. Barriers to pro-environmental behaviour are the numerous factors that hinder individuals when they try to adjust their behaviours toward living more sustainable lifestyles.

Generally, these barriers can be separated into larger categories: psychological, social/cultural, financial and structural. Psychological barriers are considered internal, where an individual's knowledge, beliefs and thoughts affect their behaviour. Social and cultural barriers are contextual, where an individual's behaviour is affected by their surroundings (e.g. neighbourhood, town, city, etc.). Financial barriers are simply a lack of funds to move toward more sustainable behaviour (e.g. new technologies, electric cars). Structural barriers are external and often impossible for an individual to control, such as lack of governmental action, or locality of residence that promotes car dependency as opposed to public transit.

Internal/psychological barriers

Further information: Media coverage of climate change

Identifying psychological barriers to pro-environmental behaviour is key to the design of successful behaviour change interventions. Scholars have identified several different categories of psychological barriers to pro-environmental action. A known researcher in the field, environmental psychologist Robert Gifford, has identified 33 of these barriers, barriers that he has termed "The Dragons of Inaction." The Dragons are separated into seven categories: Limited Cognition, Ideologies, Social Comparison, Sunk Costs, Discredence, Perceived Risks, and Limited Behaviour. Below are the seven categories, integrated with additional barriers identified by other researchers. Other psychologists have argued that the attempt to identify psychological barriers to environmental behavior is problematic when used to explain societal inaction on climate change.

Limited cognition

See also: Agnotology

Limited cognition barriers are barriers that arise from a lack of knowledge and awareness about environmental issues. For example, with a key environmental issue like climate change, a person might not engage in pro-environmental behaviour because they are: unaware that climate change is occurring; or aware that climate change is an issue, but are ill-informed about the science of climate change; or lacking information about how they could address the issue.

For those who are aware of current environmental issues, self-efficacy is an important barrier to action, where individuals often feel powerless in achieving large goals such as mitigating global climate change. Moreover, lack of motivation to change one's behaviour is correlated with the belief that individuals are incapable of performing effective pro-environmental actions.

Ideologies

Climate denial billboard

Ideological barriers are created by pre-conceived ideas and the way an individual thinks about the world. Ideologies that can create barriers to pro-environmental behaviour can include a strong belief in free-enterprise capitalism, a fatalistic belief that a higher power is in control, and a belief that technology can solve all environmental issues. Accordingly, tactics such as environmental policies have prompted a tendency to struggle against perceived threats to one's freedom and comfortable lifestyle. This barrier is namely present in Western countries where individuals enjoy comparatively high levels of objective and subjective wellbeing due to socioeconomic status. It has been noted that to live within environmental limits, there is a need to make changes to the comfortable aspects of Western lifestyles, for example, reducing meat consumption, the use of airplanes, and use of electronic gadgets with short life-spans. Western cultural norms associate meat consumption with wealth, status and luxury, and meat consumption per capita in the richest 15 nations of the world is 750% higher than in the poorest 24 nations. A shift in values may be difficult, as people's life goals are formed by their ideas of social progress, personal status, and success through careers, higher incomes and consumption.

Moreover, there exist deep structural and cultural roots that couple the macro-level of financial, property or labour institutions to the micro-level of individualistic, utilitarian values. These roots are linked to the current economic growth paradigm, which can be defined as a worldview that maintains that economic growth is both good and necessary.

Social comparison

North Hills East truck dealer. The Ford F-150 truck has been the best selling vehicle in the United States for some time achieving less than 30mpg on average.

Social comparison barriers include the comparison of actions with those of others to determine the "correct" behaviour, whether it be beneficial or harmful for the environment. This means that social comparison barriers can also facilitate pro-environmental behaviour. For example, people will alter their energy consumption to replicate the reported usage of their neighbours. Moreover, if individuals believe those around them are not actively engaging in pro-environmental behaviour, they are less likely to engage in it themselves because they believe this to be unfair.

Sunk costs

See also: Cognitive inertia

Sunk cost barriers are the investments (not necessarily financial) of an individual that in turn restrict alternative possibilities for change, or in this circumstance, for pro-environmental behaviour. One example of a financial investment is car ownership, where the individual will be less likely to use alternative modes of transportation. Habits are considered a Sunk Costs Dragon as well because they are very difficult to change (e.g. eating habits). Individuals are also deeply invested in their life goals and aspirations, even if achieving them will harm the environment. Place attachment is considered here as well, where an individual who feels no place attachment to their home will be less likely to act pro-environmentally in that place than one who loves where they live.

Additional barriers are inconvenience and time-related pressures, which are suggested as reasons why individuals go back to unsustainable habits. An individual may find it annoying and inconvenient to compost if they do not have access to municipal composting, for example, and if one is pressed for time they may choose to use their car rather than wait for public transit.

Discredence

See also: Conspiracy theories

Discredence barriers generally involve disbelief in environmental issues and/or distrust in government officials and scientists. Complete denial of climate change and other environmental issues is becoming less prominent, but it continues to persist. Skepticism is still apparent in countries where there are efforts to shape public opinion through mediums such as conservative think tanks and media outlets. Moreover, mass media is the primary source of information on climate change in many countries, therefore depending on the individual, they will either trust or ignore the information they receive which will vary from one media outlet to the next based on different views.

Distrust in government has become a prevalent issue recently. In the United States for example, Americans have been polled every year about their confidence in their country's institutions (e.g. the Supreme Court, Congress, the Presidency, and the health-care establishment), and there has been a reported collapse in trust over time (12% in 2017). From an environmental standpoint, the first Trump administration has significantly diminished regulations that were put in place by the former administration to meet environmental standards. Examples of policy changes include pulling out of the Paris Agreement, loosening regulations on toxic air pollution, and issuing an executive order that called for a 30% increase in logging on public lands. There is a 97% scientific consensus on anthropogenic climate change, yet there is still not enough being done to meet global temperature targets of staying below a 1.5 degrees Celsius increase (see Paris Agreement).

Even in a stable constitutional republic, a cynical or unmoored citizenry presents an opportunity for demagogues and populists. As much as stagnant wages in former manufacturing regions, glaring economic inequality, or white backlash after the Obama Presidency, the country's disillusionment with institutions enabled Donald Trump's election.

— The New Yorker

Perceived risk

Risk perception barriers include worrying about whether financial or temporal investments will pay off. An example of a financial investment is solar panels which are initially costly. A temporal investment can simply be spending the time to do research on the topic instead of doing something else.

There exists the concept of psychological distance, where people tend to discount future risks when making trade-offs between cost and benefits, and instead prioritize immediate day-to-day concerns. Spatial distance allows individuals to disregard any risks, and instead consider them more likely for other people and places than for themselves. This barrier can simply be thought of as "out of sight, out of mind." Additionally, people typically underestimate the likelihood of being affected by natural disasters, as well as the degree to which others are concerned about environmental issues. Furthermore, the human brain privileges experience over analysis: personal experiences with extreme weather events can influence risk perceptions, beliefs, behaviour and policy support, whereas statistical information by itself means very little to most people.

It has been hypothesised many times that no matter how strong the climate knowledge provided by risk analysts, experts and scientists is, risk perception determines agents' ultimate response in terms of mitigation. However, recent literature reports conflicting evidence about the actual impact of risk perception on agents’ climate response. Rather, a no-direct perception-response link with the mediation and moderation of many other factors and a strong dependency on the context analysed is shown. Some moderation factors considered as such in the specialised literature include communication and social norms. Yet, conflicting evidence of the disparity between public communication about climate change and the lack of behavioural change has also been observed in the general public. Likewise, doubts are raised about the observance of social norms as an influencing predominant factor that affects action on climate change. What is more, disparate evidence also showed that even agents highly engaged in mitigation (engagement is a mediation factor) actions fail ultimately to respond.

Limited behaviour

Limited behaviour barriers may include people choosing easier, yet less effective, pro-environmental behavioural changes (e.g. recycling, metal straws), and the rebound effect, which occurs when a positive environmental behaviour is followed by one that negates it (e.g. saving money with an electric car to then buy a plane ticket).

Contextual barriers

Social and cultural factors

Research has also shown that how people support and engage in pro-environmental behaviour is also affected by contextual factors (i.e. social, economic, and cultural); people with diverse cultural backgrounds have different perspectives and priorities, and thus, they may respond to the same policies and interventions in different ways with regionally differentiated world views playing an important role. This means that people will use different excuses for their behaviours depending on contextual factors. Research has shown that information has a greater impact on behaviour if it is tailored to the personal situations of consumers and resonates with their important values. This suggests that, for example, policies developed to reduce and mitigate climate change would be more effective if they were developed specifically for the people whose behaviour they were targeting.

People are social beings who respond to group norms: behaviour and decision-making has been shown to be affected by social norms and contexts.

Demographic variables like age, gender and education, can have a variety of effects on pro-environmental behaviour, depending on the issue and context. However, when considering the effects of socio-demographics on individual perceptions of climate change, a recent study reported a meta-analysis which found that the largest demographic correlation with the belief of human-caused climate change is political affiliation (e.g. conservative views often mean less support for climate mitigation).

Economic factors

The cost of sustainable alternatives and financial measures used to support new technologies can also be a barrier to pro-environmental behaviour. Households may have severe budgetary constraints that discourage them from investing in energy-efficient measures. In addition, individuals may fear that project costs will not be recovered prior to a future sale of a property. Economic factors are not just barriers to pro-environmental behaviour for individual households but are also a barrier on the international scale. Developing countries that rely on coal and fossil fuels may not have the funding or infrastructure to switch to more sustainable energy sources. Therefore, help from developed countries, with regards to cost, may be needed. As nations become more prosperous, their citizens are less concerned with the economic battle for survival and are free to pursue postmaterialistic ideals such as political freedom, personal fulfillment, and environmental conservation. In other cases however, environment-friendly behaviours may be undertaken for non-environmental reasons, such as to save money or to improve health (e.g. biking or walking instead of driving).

Structural barriers

Structural barriers are large-scale systemic barriers that may be perceived as being objective and external, and can be highly influential and near impossible to control, even when one wishes to adopt more pro-environmental behaviour. For example, lack of organizational and governmental action on sustainability is considered a barrier for individuals looking to participate in sustainable practices. Further examples of structural barriers include: low problem awareness at the local level caused by a low priority for adaptation at higher institutional levels, and missing leadership by certain key actors leading to an absence of appropriate decision-making routines. Other structural barriers reported from a Vancouver-based study include: term limits imposed on politicians that affect council's ability to make long-term decisions; budgetary cycles that force planning based on three year terms, rather than long-term planning; and hierarchical systems that inhibit flexibility and innovation.

Research has shown that individuals may not behave in accordance with environmental sustainability when they have little control over the outcome of a situation. An example of a structural choice that can influence an individual's use of high carbon transport, occurs when cities governments allow sprawling neighbourhoods to develop without associated public transit infrastructure.

The concept of barriers has also been defined in relation to adaptive capacity, the ability of a system to respond to environmental changes; a barrier can either be a reason for potential adaptive capacity not being translated into action, or a reason for the existence of low adaptive capacity.