Coherent States

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

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Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

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Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

The general states of a harmonic oscillator can be expressed as a superpostion of the energy eigenstates $|n\rangle \!$. A class of states that is of particular importance consists of the eigenstates of non-Hermitian lowering operator $a\!$, with eigenvalue $\alpha \!$:

$a|\alpha \rangle =\alpha |\alpha \rangle \!$ where $\alpha \!$ can be any complex number.

Such states are called coherent states. The term coherent reflects their important role in optics and quantum electronics. The following are some properties of coherent states.

Note that it is not possible to construct an eigenstate of $a^{\dagger }$ because $a^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle$.

I. Coherent states construction.

$|\alpha \rangle =\sum _{n=0}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}|n\rangle =e^{\alpha a^{+}}|0\rangle$

$a|\alpha \rangle =\sum _{n=0}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}a|n\rangle =\sum _{n=1}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}{\sqrt {n}}|n-1\rangle =\sum _{n=1}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {(n-1)!}}}|n-1\rangle =\alpha \left(\sum _{n=0}^{+\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}|n\rangle \right)=\alpha |\alpha \rangle$

II. Coherent states normalization.

$|\alpha \rangle =Ne^{\alpha a^{\dagger }}|0\rangle$ where $N$ is normalization constant.

$1=\langle \alpha |\alpha \rangle =\langle 0|Ne^{\alpha ^{*}a}Ne^{\alpha a^{\dagger }}|0\rangle =N^{2}\langle 0|e^{\alpha ^{*}a}e^{\alpha a^{\dagger }}|0\rangle$

For any operators A and B which both commute with their commutator, we have:

$e^{A}e^{B}=e^{A+B}e^{{\frac {1}{2}}[A,B]}$

and similarly, $e^{B}e^{A}=e^{B+A}e^{{\frac {1}{2}}[B,A]}=e^{A+B}e^{-{\frac {1}{2}}[A,B]}$

therefore: $e^{A}e^{B}=e^{B}e^{A}e^{[A,B]}\!$

Apply this result for $A=\alpha ^{*}a\!$ and $B=\alpha a^{\dagger }\!$ ( A and B both commute with their commutator because $[A,B]=|\alpha |^{2})\!$, we have:

$1=\langle \alpha |\alpha \rangle =N^{2}\langle 0|e^{\alpha ^{*}a}e^{\alpha a^{\dagger }}|0\rangle$

${\begin{aligned}N^{2}\langle 0|e^{\alpha a^{\dagger }}e^{\alpha ^{*}a}e^{[\alpha ^{*}a,\alpha a^{\dagger }]}|0\rangle &=N^{2}e^{|\alpha |^{2}}\langle 0|e^{\alpha a^{\dagger }}e^{\alpha ^{*}a}|0\rangle \\&=N^{2}e^{|\alpha |^{2}}\langle 0|e^{\alpha a^{\dagger }}|0\rangle \\&=N^{2}e^{|\alpha |^{2}}\langle 0|0\rangle \\&=N^{2}e^{|\alpha |^{2}}\end{aligned}}$

$\rightarrow N=e^{-{\frac {1}{2}}|\alpha |^{2}}$

$\rightarrow {\mbox{Normalized coherent states:}}|\alpha \rangle =e^{-{\frac {1}{2}}|\alpha |^{2}}e^{\alpha a^{\dagger }}|0\rangle$

III. Inner product of two coherent states

There is an eigenstate $|\alpha \rangle \!$ of lowering operator $a\!$ for any complex number $\alpha \!$. Therefore, we have a set of coherent states. This is NOT an orthogonal set. Indeed, the inner product of two coherent states $|\alpha \rangle \!$ and $|\beta \rangle \!$ can be calculated as follows:

${\begin{aligned}\langle \beta |\alpha \rangle &=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{-{\frac {1}{2}}|\beta |^{2}}\langle 0|e^{\beta ^{*}a}e^{\alpha a^{+}}|0\rangle \\&=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{-{\frac {1}{2}}|\beta |^{2}}\langle 0|e^{\alpha a^{+}}e^{\beta ^{*}a}e^{[\beta ^{*}a,\alpha a^{+}]}|0\rangle \\&=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{-{\frac {1}{2}}|\beta |^{2}}e^{\alpha \beta ^{*}}\langle 0|e^{\alpha a^{+}}e^{\beta ^{*}a}|0\rangle \\&=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{-{\frac {1}{2}}|\beta |^{2}}e^{\alpha \beta ^{*}}\end{aligned}}$

$\rightarrow |\langle \beta |\alpha \rangle |^{2}=e^{-|\alpha -\beta |^{2}}$

Hence, the set of coherent states is not orthogonal and the distance $|\alpha -\beta |\!$ in a complex plane measures the degree to which the two eigenstates are 'approximately orthogonal'.