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Coherent States in Quantum Mechanics
Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2
(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France
(2) Maplesoft
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Coherent states are among the most relevant representations for the state of a quantum system. These states, that form an overcomplete basis, minimize the quantum uncertainty between position x and momentum p (they satisfy the Heisenberg uncertainty principle with equality and their expectation values satisfy the classical equations of motion). Coherent states are widely used in quantum optics and quantum mechanics in general; they also mathematically characterize the concept of Planck cells. Part of this development is present in Maple 2018.2.1. To reproduce what you see below, however, you need a more recent version, as the one distributed within the Maplesoft Physics Updates (version 276 or higher). A worksheet with this contents is linked at the end of this post.
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Definition and the basics
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Set a quantum operator  and corresponding annihilation / creation operators
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![[quantumoperators = {A}]](/view.aspx?sf=210033_post/c56a3d6bc0af9fdcad1a1e09946bff22.gif)
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(1.1) |
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(1.2) |
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(1.3) |
In what follows, on the left-hand sides the product operator used is `*`, which properly represents, but does not perform the attachment of Bras Kets and operators. On the right-hand sides the product operator is `.`, that performs the attachments. Since the introduction of Physics in the Maple system, we have that
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(1.4) |
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(1.5) |
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![%Bracket(Physics:-Bra(A, n), Physics:-Ket(A, m)) = Physics:-KroneckerDelta[m, n]](/view.aspx?sf=210033_post/0fa544f732d707f0a2e584ee5d2c1ceb.gif)
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(1.6) |
New development during 2018: coherent states, the eigenstates of the annihilation operator  , with all of their properties, are now understood as such by the system
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(1.7) |
 is an eigenket of  but not of 
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(1.8) |
The norm of these states is equal to 1
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(1.9) |
These states, however, are not orthonormal as the occupation number states  are, and since  is not Hermitian, its eigenvalues are not real but complex numbers. Instead of (1.6) , we now have
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(1.10) |
At  ,
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![simplify(eval(%Bracket(Physics[Bra](`#msup(mi("a"),mo("&uminus0;"))`, alpha), Physics[Ket](`#msup(mi("a"),mo("&uminus0;"))`, beta)) = exp(-(1/2)*abs(alpha)^2-(1/2)*abs(beta)^2+conjugate(alpha)*beta), alpha = beta))](/view.aspx?sf=210033_post/9c2080f35f5a94445662fb12682b248d.gif)
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(1.11) |
Their scalar product with the occupation number states  , using the inert %Bracket on the left-hand side and the active Bracket on the other side:
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(1.12) |
The expansion of coherent states into occupation number states, first representing the product operation using `*`, then performing the attachments replacing `*` by `.`
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(1.13) |
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, Physics[Bra](A, n)), n = 0 .. infinity))*Ket(am, alpha)](/view.aspx?sf=210033_post/647ca6c5781619a811c07b114cb0ed7f.gif)
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(1.14) |
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![eval(Physics[Ket](`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Physics[`*`](Sum(Physics[`*`](Physics[Ket](A, n), Physics[Bra](A, n)), n = 0 .. infinity), Physics[Ket](`#msup(mi("a"),mo("&uminus0;"))`, alpha)), `*` = `.`)](/view.aspx?sf=210033_post/1b1359874d828ca14237256428c886ec.gif)
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(1.15) |
Hide now the ket label. When in doubt, input show to see the Kets with their labels explicitly shown
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![[hideketlabel = true]](/view.aspx?sf=210033_post/41b9475a80aefabbcb21b9101dd817cb.gif)
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(1.16) |
Define eigenkets of the annihilation operator, with two different eigenvalues for experimentation
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(1.17) |
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(1.18) |
Because the properties of coherent states are now known to the system, the following computations proceed automatically. The left-hand sides use the `*`, while the right-hand sides use the `.`
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(1.19) |
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(1.20) |
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(1.21) |
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(1.22) |
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Properties of Coherent states
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The mean value of the occupation number N
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The occupation number operator N is given by
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(2.1.1) |
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(2.1.2) |
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![value(%Dagger(Physics[`*`](`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`)) = Physics[`*`](`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`))](/view.aspx?sf=210033_post/8ddde0ec4bdbd2846d8b5fdcaea02f30.gif)
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(2.1.3) |
N is diagonal in the  basis of the Fock (occupation number) space
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![Physics:-`*`(Physics:-Bra(A, n), `#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`, Physics:-Ket(A, p)) = p*Physics:-KroneckerDelta[n, p]](/view.aspx?sf=210033_post/2d1e50fa6044b82c47f722de5f539e55.gif)
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(2.1.4) |
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The mean value of N in a coherent state 
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![Bracket(%N)[alpha] = %Bracket(Bra(am, alpha), N, Ket(am, alpha))](/view.aspx?sf=210033_post/a4a37fadec49e1afb747149c72777f59.gif)
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![Physics:-Bracket(%N)[alpha] = %Bracket(Physics:-Bra(`#msup(mi("a"),mo("&uminus0;"))`, alpha), Physics:-`*`(`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha))](/view.aspx?sf=210033_post/ae76d6b2b56a4800377d97b5b8dbf452.gif)
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(2.1.5) |
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[alpha] = %Bracket(Physics[Bra](`#msup(mi("a"),mo("&uminus0;"))`, alpha), Physics[`*`](`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`), Physics[Ket](`#msup(mi("a"),mo("&uminus0;"))`, alpha)))](/view.aspx?sf=210033_post/1f65d4e8ef18af37185b7d8518510fba.gif)
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![Physics:-Bracket(%N)[alpha] = abs(alpha)^2](/view.aspx?sf=210033_post/67a27ec52fefe27ecc3c5a4ffcf784c7.gif)
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(2.1.6) |
The mean value of 
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![Bracket(%N^2)[alpha] = %Bracket(Bra(am, alpha), N^2, Ket(am, alpha))](/view.aspx?sf=210033_post/2985144e46815b1b076a941c7961a40e.gif)
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![Physics:-Bracket(%N^2)[alpha] = %Bracket(Physics:-Bra(`#msup(mi("a"),mo("&uminus0;"))`, alpha), Physics:-`^`(Physics:-`*`(`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`), 2), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha))](/view.aspx?sf=210033_post/cb04a9025ccdbbd9a10178a759c57e3a.gif)
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(2.1.7) |
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[alpha] = %Bracket(Physics[Bra](`#msup(mi("a"),mo("&uminus0;"))`, alpha), Physics[`^`](Physics[`*`](`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`), 2), Physics[Ket](`#msup(mi("a"),mo("&uminus0;"))`, alpha)))](/view.aspx?sf=210033_post/81715f9537227c670fdf3ae70a6c3aae.gif)
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![Physics:-Bracket(%N^2)[alpha] = abs(alpha)^4+abs(alpha)^2](/view.aspx?sf=210033_post/adc36e04d5ed1754c77b8b381ecd5f21.gif)
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(2.1.8) |
The standard deviation ![`ΔN` = sqrt(-Bracket(%N)[alpha]^2+Bracket(%N^2)[alpha])](/view.aspx?sf=210033_post/18553d872bf3a952fd90feca5f90eff5.gif) for a state 
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[alpha] = abs(alpha)^4+abs(alpha)^2)-(Physics[Bracket](%N)[alpha] = abs(alpha)^2)^2)^(1/2)](/view.aspx?sf=210033_post/19b43afffd566c595453ea662a65bd97.gif)
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![(-Physics:-Bracket(%N)[alpha]^2+Physics:-Bracket(%N^2)[alpha])^(1/2) = abs(alpha)](/view.aspx?sf=210033_post/48fa4b3388b806052741db6701ed7884.gif)
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(2.1.9) |
In conclusion, a coherent state  has a finite spreading  . Coherent states are good approximations for the states of a laser, where the laser intensity I is proportional to the mean value of the photon number, I f ![Bracket(%N)[alpha] = abs(alpha)^2](/view.aspx?sf=210033_post/5fd4c86e4c7c35a4ff6eb69a1634119e.gif) , and so the intensity fluctuation,  .
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The mean value of the occupation number N in an occupation number state 
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![Bracket(%N)[n] = %Bracket(Bra(A, n), N, Ket(A, n))](/view.aspx?sf=210033_post/4d8060bf2f4292d2e4fe17c3a262f7ea.gif)
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![Physics:-Bracket(%N)[n] = %Bracket(Physics:-Bra(A, n), Physics:-`*`(`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`), Physics:-Ket(A, n))](/view.aspx?sf=210033_post/40eeb0120cbdd321a1411796bb73ff3a.gif)
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(2.1.10) |
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[n] = %Bracket(Bra(A, n), Physics[`*`](`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`), Ket(A, n)))](/view.aspx?sf=210033_post/cd351c28a8045fc102d6094a39577191.gif)
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![Physics:-Bracket(%N)[n] = n](/view.aspx?sf=210033_post/5a452d25235e5565d51d292a91bc7e09.gif)
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(2.1.11) |
The mean value of the occupation number N in a state  is thus n itself, as expected since  represents a (Fock space) state of n (quase-) particles. Accordingly,
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![Bracket(%N^2)[n] = %Bracket(Bra(A, n), N^2, Ket(A, n))](/view.aspx?sf=210033_post/a11107f22ba53cf4e07eb33e04289fa0.gif)
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![Physics:-Bracket(%N^2)[n] = %Bracket(Physics:-Bra(A, n), Physics:-`^`(Physics:-`*`(`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`), 2), Physics:-Ket(A, n))](/view.aspx?sf=210033_post/bd0a5fb10bf33231e18b3cb90b2073b9.gif)
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(2.1.12) |
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[n] = %Bracket(Bra(A, n), Physics[`^`](Physics[`*`](`#msup(mi("a"),mo("+"))`, `#msup(mi("a"),mo("&uminus0;"))`), 2), Ket(A, n)))](/view.aspx?sf=210033_post/fcb837a41e0e3bebc53b238c4d84c845.gif)
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![Physics:-Bracket(%N^2)[n] = n^2](/view.aspx?sf=210033_post/8f06c6faa887458c84d3e0a90839d290.gif)
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(2.1.13) |
The standard deviation ![`ΔN` = sqrt(-Bracket(%N)[n]^2+Bracket(%N^2)[n])](/view.aspx?sf=210033_post/21ab8cfebdd067d51e11b166e12e1aa0.gif) for a state  , is thus
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[n] = n^2)-(Physics[Bracket](%N)[n] = n)^2)^(1/2)](/view.aspx?sf=210033_post/ab131ecc9fcaf8644d46431ba1813bfe.gif)
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![(-Physics:-Bracket(%N)[n]^2+Physics:-Bracket(%N^2)[n])^(1/2) = 0](/view.aspx?sf=210033_post/2ffdc8e0f44030792a4d6bcf65cd6c6b.gif)
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(2.1.14) |
That is, in a Fock state,  , there is no intensity fluctuation.
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The specific properties of coherent states implemented can be derived explicitly departing from the projection of  into the  basis of occupation number states and the definition of  as the operator that annihilates the vacuum  
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, Bra(A, n)), n = 0 .. infinity))*Ket(am, alpha)](/view.aspx?sf=210033_post/fffbd243aeb971f39211f614bf206617.gif)
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(2.2.1) |
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, Bra(A, n)), n = 0 .. infinity), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)), `*` = `.`)](/view.aspx?sf=210033_post/50f670ba5143d78e611d25b2b4302b1d.gif)
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(2.2.2) |
To derive  from the formula above, start multiplying by 
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(2.2.3) |
In view of  , discard the first term of the sum
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![subs(0 = 1, Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity)))](/view.aspx?sf=210033_post/4cd12d1a47c32930d845cfcbcbe496fc.gif)
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(2.2.4) |
Change variables  ; in the result rename 
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![subs(k = n, PDEtools:-dchange(n = k+1, Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Ket(A, n)/factorial(n)^(1/2), n = 1 .. infinity)), `@`(combine, simplify)))](/view.aspx?sf=210033_post/59ab97e99c31e46045f2c50a4b4dcbce.gif)
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(2.2.5) |
Activate the product  by replacing, in the right-hand side, the product operator `*` by `.`
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![lhs(Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Sum(exp(-(1/2)*abs(alpha)^2)*Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Ket(A, n+1))*alpha^(n+1)/(factorial(n)^(1/2)*(n+1)^(1/2)), n = 0 .. infinity)) = eval(rhs(Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Sum(exp(-(1/2)*abs(alpha)^2)*Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Ket(A, n+1))*alpha^(n+1)/(factorial(n)^(1/2)*(n+1)^(1/2)), n = 0 .. infinity)), `*` = `.`)](/view.aspx?sf=210033_post/686aace5d89c9314f5272f9dc558ab12.gif)
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(2.2.6) |
By inspection the right-hand side of (2.2.6) is equal to  times the right-hand side of (2.2.2)
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![alpha*(Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity))-(Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Sum(exp(-(1/2)*abs(alpha)^2)*Ket(A, n)*alpha^(n+1)/factorial(n)^(1/2), n = 0 .. infinity))](/view.aspx?sf=210033_post/8a30a27b45be368814f0af26412d9c84.gif)
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(2.2.7) |
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![combine(alpha*Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)-Physics[`*`](`#msup(mi("a"),mo("&uminus0;"))`, Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = alpha*(Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity))-(Sum(exp(-(1/2)*abs(alpha)^2)*Ket(A, n)*alpha^(n+1)/factorial(n)^(1/2), n = 0 .. infinity)))](/view.aspx?sf=210033_post/ee0dbcdf25620e55e2f21a8efd4935da.gif)
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(2.2.8) |
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Overview of the coherent states distribution
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Consider the projection of  over an occupation number state 
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, Bra(A, n)), n = 0 .. infinity), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)))) = Bracket(Bra(A, n), rhs(Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Physics[`*`](Sum(Physics[`*`](Ket(A, n), Bra(A, n)), n = 0 .. infinity), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha))))](/view.aspx?sf=210033_post/2b461446bac3497945cff10fb9628d57.gif)
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(2.2.9) |
An overview of the distribution of coherent states  for a sample of values of n and  is thus as follows
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The distribution can be explored for ranges of values of n and  using Explore
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![Explore(plot(rhs(%Bracket(Bra(A, n), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = exp(-(1/2)*abs(alpha)^2)*alpha^n/factorial(n)^(1/2)), n = 0 .. 200, view = 0 .. .6, labels = [n, NA]), parameters = [alpha = 0 .. 10], initialvalues = [alpha = 5])](/view.aspx?sf=210033_post/45915be0eebbfba260ea963b0ee375be.gif)
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The identity in the title can be derived departing again from the the projection of a coherent state into the  basis of occupation number states
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(2.6.1) |
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(2.6.2) |
Taking the `*` product of these two expressions
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(2.6.3) |
Perform the attachment of Bras and Kets on the right-hand side by replacing `*` by `.`, evaluating the sum and simplifying the result
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![lhs(Physics[`*`](Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Physics[`*`](Sum(exp(-(1/2)*abs(beta)^2)*conjugate(beta)^k*Bra(A, k)/factorial(k)^(1/2), k = 0 .. infinity), Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity))) = simplify(value(eval(rhs(Physics[`*`](Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Physics[`*`](Sum(exp(-(1/2)*abs(beta)^2)*conjugate(beta)^k*Bra(A, k)/factorial(k)^(1/2), k = 0 .. infinity), Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity))), `*` = `.`)))](/view.aspx?sf=210033_post/3e835e2fd57568661c6e188ee57563fb.gif)
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(2.6.4) |
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Overview of the real and imaginary part of 
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In most cases,  and  are complex valued numbers. Below, the plots assume that  and  are both real. To take into account the general case, the possibility to tune a phase difference  between  and  is explicitly added, so that (2.6.4) becomes
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![%Bracket(Bra(am, beta), Ket(am, alpha)) = subs(conjugate(beta) = conjugate(beta)*exp(I*theta), rhs(Physics[`*`](Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = exp(conjugate(beta)*alpha-(1/2)*abs(beta)^2-(1/2)*abs(alpha)^2)))](/view.aspx?sf=210033_post/da85464d093b62bad5d05ea2f463f12e.gif)
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(2.6.5) |
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![Explore(plot3d(Re(rhs(%Bracket(Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = exp(-(1/2)*abs(beta)^2-(1/2)*abs(alpha)^2+alpha*conjugate(beta)*exp(I*theta)))), alpha = -10 .. 10, beta = -10 .. 10, view = -1 .. 1, orientation = [-12, 74, 3], axes = boxed), parameters = [theta = 0 .. 2*Pi], initialvalues = [theta = (1/10)*Pi])](/view.aspx?sf=210033_post/0cf4a49e95d6dce5c959d0e773bb823c.gif)
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