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The general states of a harmonic oscillator can be expressed as a superpostion of the energy eigenstates <math>|n\rangle\!</math>. A class of states that is of particular importance consists of the eigenstates of non-Hermitian lowering operator <math>a\!</math>, with eigenvalue <math>\alpha\!</math>:
{{Quantum Mechanics A}}
The general states of a [[Harmonic Oscillator Spectrum and Eigenstates|harmonic oscillator]] can be expressed as a superpostion of the energy eigenstates <math>|n\rangle.\!</math> A class of states that is of particular importance are the eigenstates of the (non-Hermitian) lowering operator <math>\hat{a},\!</math>
 
<math>a|\alpha\rangle=\alpha|\alpha\rangle,\!</math>


<math>a|\alpha\rangle=\alpha|\alpha\rangle\!</math>
where <math>\alpha\!</math> can be any complex number.
where <math>\alpha\!</math> can be any complex number.
   
   
Such states are called coherent states. The term coherent reflects their important role in optics and quantum electronics.  
These states are known as coherent states. The term, "coherent", reflects their important role in optics and quantum electronics.
Note that it is not possible to construct an eigenstate of the raising operator <math>\hat{a}^{\dagger}</math> because
<math>a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle;</math> this fact means that application of <math>\hat{a}^{\dagger}</math> to any superposition of harmonic oscillator eigenstates eliminates the lowest-energy state that was present in the superposition.
 
The following are some properties of coherent states.
The following are some properties of coherent states.
   
   
Note that it is not possible to construct an eigenstate of <math>a^{\dagger}</math> because
== Construction of Coherent States ==
<math>a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle</math>.
 
The coherent state with eigenvalue <math>\alpha\!</math> is given by
I. Coherent states construction.
 
:<math>|\alpha\rangle=\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle=e^{\alpha\hat{a}^\dagger}|0\rangle.</math>


:<math>|\alpha\rangle=\sum_{n=0}^{+\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle=e^{\alpha a^+}|0\rangle</math>
We may see that this is a coherent state with the given eigenvalue as follows:


:<math>a|\alpha\rangle=\sum_{n=0}^{+\infty}\frac{\alpha^n}{\sqrt{n!}}a|n\rangle=\sum_{n=1}^{+\infty}
:<math>\hat{a}|\alpha\rangle=\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\hat{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=
\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</math>
\alpha\left(\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle\right)=\alpha|\alpha\rangle</math>
   
   
II. Coherent states normalization.
This state, however, is not normalized, so we will now normalize it. Let us introduce a normalization constant, <math>N,\!</math> into the coherent state:
   
   
<math>|\alpha\rangle=Ne^{\alpha a^{\dagger}} |0\rangle </math>
<math>|\alpha\rangle=Ne^{\alpha a^{\dagger}} |0\rangle </math>
where <math>N</math> is normalization constant.


:<math>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 </math>
We now determine what value of <math>N\!</math> yields a normalized state:


For any operators A and B which both commute with their commutator, we have:  
<math>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 </math>
 
We now use the fact that, for any two operators <math>\hat{A}</math> and <math>\hat{B}</math> that both commute with their commutator, the following formula, known as the Campbell-Baker-Hausdorff formula, holds:
                                                  
                                                  
<math>e^A e^B = e^{A+B} e^{\frac{1}{2}[A,B]} </math>
<math>e^\hat{A} e^\hat{B} = e^{\hat{A}+\hat{B}} e^{[\hat{A},\hat{B}]/2}\!</math>


Similarly,


and similarly,  <math>e^B e^A = e^{B+A} e^{\frac{1}{2}[B,A]} = e^{A+B} e^{-\frac{1}{2}[A,B]}</math>
<math>e^\hat{B} e^\hat{A} = e^{\hat{B}+\hat{A}} e^{[\hat{B},\hat{A}]/2} = e^{\hat{A}+\hat{B}} e^{-[\hat{A},\hat{B}]/2}.</math>


Combining the above two formulas, we obtain


therefore:    <math>e^A e^B = e^B e^A e^{[A,B]}\!</math>
<math>e^\hat{A} e^\hat{B} = e^\hat{B} e^\hat{A} e^{[\hat{A},\hat{B}]}.\!</math>


Apply this result for <math>A=\alpha ^* a\!</math> and <math>B=\alpha a^{\dagger}\!</math> ( A and B both commute with their commutator because
This result applies for <math>\hat{A}=\alpha^\ast \hat{a}\!</math> and <math>\hat{B}=\alpha \hat{a}^{\dagger}\!</math> because the commutator for these two operators is <math>[\hat{A},\hat{B}]=|\alpha|^2,\!</math>, which is a constant.  We thus obtain
<math>[A,B]=|\alpha|^2)\!</math>, we have:


:<math>1=\langle\alpha|\alpha\rangle = N^2\langle 0|e^{\alpha^*a} e^{\alpha a^{\dagger}} |0\rangle </math>
:<math>\begin{align}
 
\langle\alpha|\alpha\rangle &= N^2\langle 0|e^{\alpha^\ast\hat{a}}e^{\alpha\hat{a}^{\dagger}}|0\rangle=N^2\langle 0|e^{\alpha\hat{a}^{\dagger}}e^{\alpha^\ast\hat{a}}e^{[\alpha^\ast\hat{a},\alpha\hat{a}^{\dagger}]} |0\rangle \\
:<math>
&=N^2e^{|\alpha|^2}\langle 0|e^{\alpha\hat{a}^{\dagger}}e^{\alpha^\ast\hat{a}}|0\rangle=N^2e^{|\alpha|^2}\langle 0|e^{\alpha\hat{a}^{\dagger}} |0\rangle \\  
\begin{align}
&=N^2e^{|\alpha|^2}\langle 0|0\rangle=N^2e^{|\alpha|^2}
N^2\langle 0|e^{\alpha a^{\dagger}} e^{\alpha^* a} e^{[\alpha^*a,\alpha a^{\dagger}]} |0\rangle
&= N^2e^{|\alpha|^2}\langle 0|e^{\alpha a^{\dagger}} e^{\alpha^* a} |0\rangle \\
&= N^2e^{|\alpha|^2}\langle 0|e^{\alpha a^{\dagger}} |0\rangle \\  
&= N^2e^{|\alpha|^2}\langle 0|0\rangle \\
&= N^2e^{|\alpha|^2}
\end{align}
\end{align}
</math>
</math>


:<math>\rightarrow N=e^{-\frac{1}{2}|\alpha|^2}</math>
We have thus determined the normalization constant,
 
<math>\rightarrow N=e^{-\frac{1}{2}|\alpha|^2}.</math>
 
The normalized coherent state <math>|\alpha\rangle</math> is therefore


:<math>\rightarrow \mbox{Normalized coherent states:} |\alpha \rangle = e^{-\frac{1}{2}|\alpha |^2 } e^ {\alpha a^{\dagger} }|0 \rangle </math>
<math>|\alpha\rangle=e^{-|\alpha |^2/2}e^{\alpha\hat{a}^{\dagger}}|0\rangle.</math>
   
   
III. Inner product of two coherent states
== Inner Product of Two Coherent States ==
   
   
There is an eigenstate <math>|\alpha\rangle\!</math>  
We have shown that, for any complex number <math>\alpha,\!</math> there is an eigenstate <math>|\alpha\rangle\!</math>  
of lowering operator <math>a\!</math> for any complex number <math>\alpha\!</math>. Therefore, we have a set of coherent states. This is NOT an orthogonal set.
of the lowering operator <math>\hat{a}.</math> Therefore, we have a complete set of coherent states. However, this is ''not'' an orthogonal set. Indeed, the inner product of two coherent states <math>|\alpha\rangle\!</math> and <math>|\beta\rangle\!</math> can be calculated as follows:
Indeed, the inner product of two coherent states <math>|\alpha\rangle\!</math> and <math>|\beta\rangle\!</math> can be calculated as follows:


:<math>
<math>
\begin{align}
\begin{align}
\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 \\
\langle \beta|\alpha \rangle &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}\langle 0|e^{\beta^\ast\hat{a}} e^{\alpha\hat{a}^\dagger} |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}\langle 0|e^{\alpha\hat{a}^\dagger} e^{\beta^\ast\hat{a}} e^{[\beta^\ast\hat{a},\alpha\hat{a}^\dagger]}|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^*}\langle 0|e^{\alpha\hat{a}^\dagger} e^{\beta^\ast\hat{a}}|0\rangle \\
&= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}e^{\alpha \beta^*}  
&= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}e^{\alpha \beta^*}
\end{align}
\end{align}
</math>
</math>


<math>\rightarrow |\langle \beta|\alpha \rangle |^2 = e^{-|\alpha-\beta|^2}</math>
<math>|\langle\beta|\alpha\rangle |^2 = e^{-|\alpha-\beta|^2}.</math>


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

Latest revision as of 13:43, 12 August 2013

Quantum Mechanics A
SchrodEq.png
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle} evolves in time.
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
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
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
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
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n\rangle.\!} A class of states that is of particular importance are the eigenstates of the (non-Hermitian) lowering operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a},\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a|\alpha\rangle=\alpha|\alpha\rangle,\!}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\!} can be any complex number.

These states are known as coherent states. The term, "coherent", reflects their important role in optics and quantum electronics.

Note that it is not possible to construct an eigenstate of the raising operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}^{\dagger}} because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle;} this fact means that application of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}^{\dagger}} to any superposition of harmonic oscillator eigenstates eliminates the lowest-energy state that was present in the superposition.

The following are some properties of coherent states.

Construction of Coherent States

The coherent state with eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\!} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle=\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle=e^{\alpha\hat{a}^\dagger}|0\rangle.}

We may see that this is a coherent state with the given eigenvalue as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}|\alpha\rangle=\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\hat{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}

This state, however, is not normalized, so we will now normalize it. Let us introduce a normalization constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N,\!} into the coherent state:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle=Ne^{\alpha a^{\dagger}} |0\rangle }

We now determine what value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\!} yields a normalized state:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 }

We now use the fact that, for any two operators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{B}} that both commute with their commutator, the following formula, known as the Campbell-Baker-Hausdorff formula, holds:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^\hat{A} e^\hat{B} = e^{\hat{A}+\hat{B}} e^{[\hat{A},\hat{B}]/2}\!}

Similarly,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^\hat{B} e^\hat{A} = e^{\hat{B}+\hat{A}} e^{[\hat{B},\hat{A}]/2} = e^{\hat{A}+\hat{B}} e^{-[\hat{A},\hat{B}]/2}.}

Combining the above two formulas, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^\hat{A} e^\hat{B} = e^\hat{B} e^\hat{A} e^{[\hat{A},\hat{B}]}.\!}

This result applies for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}=\alpha^\ast \hat{a}\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{B}=\alpha \hat{a}^{\dagger}\!} because the commutator for these two operators is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{A},\hat{B}]=|\alpha|^2,\!} , which is a constant. We thus obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \langle\alpha|\alpha\rangle &= N^2\langle 0|e^{\alpha^\ast\hat{a}}e^{\alpha\hat{a}^{\dagger}}|0\rangle=N^2\langle 0|e^{\alpha\hat{a}^{\dagger}}e^{\alpha^\ast\hat{a}}e^{[\alpha^\ast\hat{a},\alpha\hat{a}^{\dagger}]} |0\rangle \\ &=N^2e^{|\alpha|^2}\langle 0|e^{\alpha\hat{a}^{\dagger}}e^{\alpha^\ast\hat{a}}|0\rangle=N^2e^{|\alpha|^2}\langle 0|e^{\alpha\hat{a}^{\dagger}} |0\rangle \\ &=N^2e^{|\alpha|^2}\langle 0|0\rangle=N^2e^{|\alpha|^2} \end{align} }

We have thus determined the normalization constant,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rightarrow N=e^{-\frac{1}{2}|\alpha|^2}.}

The normalized coherent state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle} is therefore

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle=e^{-|\alpha |^2/2}e^{\alpha\hat{a}^{\dagger}}|0\rangle.}

Inner Product of Two Coherent States

We have shown that, for any complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha,\!} there is an eigenstate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle\!} of the lowering operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}.} Therefore, we have a complete set of coherent states. However, this is not an orthogonal set. Indeed, the inner product of two coherent states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\beta\rangle\!} can be calculated as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \langle \beta|\alpha \rangle &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}\langle 0|e^{\beta^\ast\hat{a}} e^{\alpha\hat{a}^\dagger} |0\rangle \\ &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}\langle 0|e^{\alpha\hat{a}^\dagger} e^{\beta^\ast\hat{a}} e^{[\beta^\ast\hat{a},\alpha\hat{a}^\dagger]}|0\rangle \\ &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}e^{\alpha \beta^*}\langle 0|e^{\alpha\hat{a}^\dagger} e^{\beta^\ast\hat{a}}|0\rangle \\ &= e^{-\frac{1}{2}|\alpha|^2}e^{-\frac{1}{2}|\beta|^2}e^{\alpha \beta^*} \end{align} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\langle\beta|\alpha\rangle |^2 = e^{-|\alpha-\beta|^2}.}

Hence, the set of coherent states is not orthogonal and the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha-\beta|\!} in the complex plane measures the degree to which the two eigenstates are "approximately orthogonal".

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