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u u ψ ψ ψ 1 displaystyle langle psi psi rangle langle psi hat u dagger hat u psi rangle langle psi psi rangle 1 in this ideal case all the energy impinging on the crystal emerges from the crystal an operator u with the property that u u i displaystyle hat u dagger 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 edit 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 u i i h displaystyle hat u approx i i hat h and the adjoint by u i i h displaystyle hat u dagger approx i i hat h dagger energy conservation then requires i u u i i h i i h i i h i h displaystyle i hat u dagger hat u approx left i i hat h dagger right left i i hat h right approx i i hat h dagger i hat h this requires that h h displaystyle hat h hat h dagger operators like this that are equal to their adjoints are called hermitian or self adjoint the infinitesimal transition of the polarization state is ψ ψ i h ψ displaystyle psi rangle psi rangle i hat h psi rangle thus energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a hermitian operator photons connection to quantum mechanics edit main article photon energy momentum and angular momentum of photons edit energy edit 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 ϵ ℏ ω displaystyle epsilon hbar omega where ℏ displaystyle hbar is an experimentally determined quantity known as the reduced planck constant if there are n displaystyle n photons in a box of volume v displaystyle v the energy in the electromagnetic field is n ℏ ω displaystyle n hbar omega and the energy density is n ℏ ω v displaystyle n hbar omega over 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 n displaystyle n the quantum energy density must be the same as the classical energy density n ℏ ω v e c e 2 8 π displaystyle n hbar omega over v mathcal e _ c frac vert mathbf e vert 2 8 pi the number of photons in the box is then n v 8 π ℏ ω e 2 displaystyle n frac v 8 pi hbar omega vert mathbf e vert 2 momentum edit the correspondence principle also determines the momentum and angular momentum of the photon for momentum p z n ℏ ω c v n ℏ k z v displaystyle mathcal p _ z n hbar omega over cv n hbar k_ z over v where k z displaystyle k_ z is the wave number this implies that the momentum of a photon is p z ℏ k z displaystyle p_ z hbar k_ z angular momentum and spin edit similarly for the spin angular momentum l 1 ω e c ψ r 2 ψ l 2 n ℏ v ψ r 2 ψ l 2 displaystyle mathcal l frac 1 omega mathcal e _ c left vert psi _ rm r vert 2 vert psi _ rm l vert 2 right frac n hbar v left vert psi _ rm r vert 2 vert psi _ rm l vert 2 right where e c displaystyle mathcal e _ c is field strength this implies that the spin angular momentum of the photon is l z ℏ ψ r 2 ψ l 2 displaystyle l_ z hbar left vert psi _ rm r vert 2 vert psi _ rm l vert 2 right the quantum interpretation of this expression is that the photon has a probability of ψ r 2 displaystyle mid psi _ rm r mid 2 of having a spin angular momentum of ℏ displaystyle hbar and a probability of ψ l 2 displaystyle mid psi _ rm l mid 2 of having a spin angular momentum of ℏ displaystyle hbar 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 2 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 edit main article spin angular momentum of light the spin of the photon is defined as the coefficient of ℏ displaystyle hbar in the spin angular momentum calculation a photon has spin 1 if it is in the r displaystyle r rangle state and 1 if it is in the l displaystyle l rangle state the spin operator is defined as the outer product s d e f r r l l 0 i i 0 displaystyle hat s stackrel mathrm def mathrm r rangle langle mathrm r mathrm l rangle langle mathrm l begin pmatrix 0 i i 0 end pmatrix the eigenvectors of the spin operator are r displaystyle mathrm r rangle and l displaystyle mathrm l rangle 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 citation needed the expected value of a spin measurement on a photon is then ψ s ψ ψ r 2 ψ l 2 displaystyle langle psi hat s psi rangle vert psi _ rm r vert 2 vert psi _ rm l vert 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 edit see also circular polarization quantum mechanics we can write the circularly polarized states as s displaystyle s rangle where s 1 for r displaystyle mathrm r rangle and s 1 for l displaystyle mathrm l rangle an arbitrary state can be written ψ s 1 1 a s exp i α x i s θ s displaystyle psi rangle sum _ s 1 1 a_ s exp left i alpha _ x is theta right s rangle where α 1 displaystyle alpha _ 1 and α 1 displaystyle alpha _ 1 are phase angles θ is the angle by which the frame of reference is rotated and s 1 1 a s 2 1 displaystyle sum _ s 1 1 vert a_ s vert 2 1 spin and angular momentum operators in differential form edit when the state is written in spin notation the spin operator can be written s d i θ displaystyle hat s _ d rightarrow i partial over partial theta s d i θ displaystyle hat s _ d dagger rightarrow i partial over partial theta the eigenvectors of the differential spin operator are exp i α x i s θ s displaystyle exp left i alpha _ x is theta right s rangle to see this note s d exp i α x i s θ s i θ exp i α x i s θ s s exp i α x i s θ s displaystyle hat s _ d exp left i alpha _ x is theta right s rangle rightarrow i partial over partial theta exp left i alpha _ x is theta right s rangle s left exp left i alpha _ x is theta right s rangle right the spin angular momentum operator is l z ℏ s d displaystyle hat l _ z hbar hat s _ d nature of probability in quantum mechanics edit probability for a single photon edit 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 edit 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 clarification needed 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 r x y r displaystyle langle r x rangle langle y r rangle the product of the individual amplitudes 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 y r r x displaystyle langle y r rangle langle r x rangle plus the amplitude for it to pass as a left circularly polarized photon y l l x displaystyle langle y l rangle langle l x rangle dots the total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2 uncertainty principle edit main article uncertainty principle cauchy schwarz inequality in euclidean space v w v w cos a displaystyle mathbf v cdot mathbf w left mathbf v right left mathbf w right cos a this implies v w v w displaystyle mathbf v cdot mathbf w leq left mathbf v right left mathbf w right mathematical preparation edit for any legal clarification needed operators the following inequality a consequence of the cauchy schwarz inequality is true 1 4 a b b a x x 2 a x 2 b x 2 displaystyle frac 1 4 left langle hat a hat b hat b hat a x x rangle right 2 leq left hat a x right 2 left hat b x right 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 δ ψ a δ ψ b 1 2 a b ψ displaystyle delta _ psi hat a delta _ psi hat b geq frac 1 2 left left langle left hat a hat b right right rangle _ psi right where x ψ ψ x ψ displaystyle left langle hat x right rangle _ psi left langle psi right hat x left psi right rangle is the operator mean of observable x in the system state ψ and δ ψ x x 2 ψ x ψ 2 displaystyle delta _ psi hat x sqrt langle hat x 2 rangle _ psi langle hat x rangle _ psi 2 here a b d e f a b b a displaystyle left hat a hat b right stackrel mathrm def 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 edit 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 δ ψ l z δ ψ θ ℏ 2 displaystyle delta _ psi hat l _ z delta _ psi theta geq frac hbar 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 edit 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 le...