91Ó°ÊÓ

Coherent states of the harmonic oscillator. Among the stationary states of the harmonic oscillator (Equation 2.67) only n = 0 hits the uncertainty limit $\left({\mathrm{σ}}_x{\mathrm{σ}}_p=\sout{h}/2\right)$; in general, ${\mathbf{σ}}_{\mathbf{x}}{\mathbf{σ}}_{\mathbf{p}}\mathbf{=}\left(2n+1\right)\sout{\mathbf{h}}\mathbf{}\mathbf{/}\mathbf{2}$, as you found in Problem 2.12. But certain linear combinations (known as coherent states) also minimize the uncertainty product. They are (as it turns out) Eigen functions of the lowering operator

${\mathit{ψ}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{n}\mathbf{!}}}{\left({\hat{a}}_{+}\right)}^{\mathbf{n}}{\mathit{ψ}}_{\mathbf{0}}$(2.68).

$\mathit{a}\mathit{\_}\left|\mathrm{Î\pm }>=\mathrm{Î\pm }\right|\mathit{a}\mathbf{>}$(the Eigen value α can be any complex number).

(a)Calculate $<x>\mathbf{,}<x^2>\mathbf{,}<p>\mathbf{,}<p^2>$in the state |α〉. Hint: Use the technique in Example 2.5, and remember that is the Hermitian conjugate of $\mathit{a}\mathbf{-}$. Do not assume α is real.

(b) Find ${\mathit{σ}}_{\mathbf{x}}$; show that ${\mathit{σ}}_{\mathbf{x}}{\mathit{σ}}_{\mathbf{p}}\mathbf{=}\sout{\mathbf{h}}\mathbf{}\mathbf{/}\mathbf{2}$.

(c) Like any other wave function, a coherent state can be expanded in terms of energy Eigen states: $\left|\mathrm{Î\pm }>=\underset{n=0}{\overset{∞}{∑}}{C}_n\right|\mathbf{}\mathit{n}\mathbf{>}\mathbf{.}$

Show that the expansion coefficients are${\mathit{c}}_{\mathbf{n}}\mathbf{=}\frac{{\mathbf{Î\pm }}^{\mathbf{n}}}{\sqrt{\mathbf{n}\mathbf{!}}}{\mathit{c}}_{\mathbf{0}}\mathbf{.}$

(d) Determine by normalizing |α〉. Answer: exp$\left(-{\left|\mathrm{Î\pm }\right|}^2/2\right)$

(e) Now put in the time dependence: $\left|n>→e^{-i{E}_{n}tI\sout{h}}\right|\mathbf{}\mathit{n}\mathbf{>}$,

and show that $\left|\mathrm{Î\pm }\left(t\right)\right|$remains an Eigen state of $\mathit{a}\mathbf{-}$, but the Eigen value evolves in time:$\mathit{Î\pm }\left(t\right)\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{Ó\neg }\mathbf{t}}$ So a coherent state stays coherent, and continues to minimize the uncertainty product.

(f) Is the ground state $\left(\left|n=0>\right|\right)$itself a coherent state? If so, what is the Eigen value?

Step by step solution

01

(a) Calculate <x>,<x2>,<p>,<p2> in the state |α〉.

02

 Step2: (b) Finding σx and σp

03

(c) Showing the expansion co-efficients.

Using Eq. 2.67 for ${\mathrm{ψ}}_n$:

${\mathrm{ψ}}_n=\frac{1}{\sqrt{n!}}{\left({\hat{a}}_{+}\right)}^n{\mathrm{ψ}}_0$ (2.68).

04

(d) Determine by normalizing |α〉.

$1=\underset{n=0}{\overset{∞}{∑}}{\left|c_n\right|}^2={\left|c_0\right|}^2\underset{n=0}{\overset{∞}{∑}}\frac{{\left|\mathrm{Î\pm }\right|}^{2n}}{n!}={\left|c_0\right|}^2{e}^{{\left|a\right|}^2}⇒c_0=e^{-{\left|a\right|}^2/2}.$

05

 Step5:(e) showing that |αt|remains an Eigen state of a- 

$|\mathrm{Î\pm }\left(t\right)>=\underset{n=0}{\overset{∞}{∑}}{c}_n{e}^{-i{E}_{n}tI\sout{h}}|n>=\underset{n=0}{\overset{∞}{∑}}\frac{{\mathrm{Î\pm }}^n}{\sqrt{n!}}{e}^{-{\left|\mathrm{Î\pm }\right|}^2/2}{e}^{-i\left(n+\frac{1}{2}\right)\mathrm{Ó\neg }t}|n>=e^{-i\mathrm{Ó\neg }t/2}\underset{n=0}{\overset{∞}{∑}}\frac{{\left(\mathrm{Î\pm }{e}^{-i\mathrm{Ó\neg }t}\right)}^n}{\sqrt{n!}}{e}^{-{\left|\mathrm{Î\pm }\right|}^2/2}|n>$

Apart from the overall phase factor $e^{-i\mathrm{Ó\neg }t/2}$ (which doesn’t affect its status as an Eigen function of a_-a− ,

or its Eigen value), $|\mathrm{Î\pm }\left(t\right)>$is the same as$|\mathrm{Î\pm }>$, but with eigenvalue$\mathrm{Î\pm }\left(t\right)=e^{-i\mathrm{Ó\neg }t}\mathrm{Î\pm }.$

06

(f) Eigen value.

Equation 2.59 says $a_{-}|{\mathrm{ψ}}_0>=0$, so yes, it is a coherent state, with Eigen value α = 0.

${\hat{a}}_{-}{\mathrm{ψ}}_0=0$ (2.59).

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