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Monday, November 11, 2013

Momentum-Space Wavefunction - Expectation Values: Quantum Mechanics (Griffiths)

In D. J. Griffiths' "Introduction to Quantum Mechanics: 1st Edition," Problem 3.51 on page 117 is a 3-star problem. In the second edition, it's problem 3.12 on page 109, and is not a starred problem at all. I attribute this to being given more information about \Psi_{x,t} in terms of \Phi_{p,t} in the second edition. In the first edition, we are working from knowledge of the form of \Phi alone. Since one is a Fourier transform of the other and vice versa, we can convert the integral used to evaluate < x>  from one with an integrand function containing \Psi and x to one containing \Phi and the x operator.

< x> = \int \Phi^{*}\left ( -\frac{\hbar}{i}\frac{\partial }{\partial p} \right )\Phi dp

In the first edition, we're given

\Phi(p,t)=\frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^{\infty}e^{-ipx/\hbar}\Psi(x,t)dx        (1)

We know <x>=\int_{-\infty}^{\infty}x|\Psi(x,t)|^{2}dx, so we just need to show that

\int \Phi^{*}\left ( -\frac{\hbar}{i}\frac{\partial }{\partial p} \right )\Phi dp=\int x|\Psi(x,t)|^{2}dx

Easier said than done, but here we go. It's not too bad, actually...

\int \Phi^{*}\left ( -\frac{\hbar}{i}\frac{\partial }{\partial p} \right )\Phi dp

Using (1) gives

 \int \left [ \frac{1}{\sqrt{2\pi \hbar}}\int e^{ipx^{'}/\hbar}\Psi^{*}dx^{'} \right ]\left ( -\frac{\hbar}{i} \frac{\partial }{\partial p}\right )\left [ \frac{1}{\sqrt{2\pi \hbar}}\int e^{-ipx/\hbar}\Psi dx \right ]dp

Simplifying a little...

\int \left [ \frac{1}{\sqrt{2\pi \hbar}}\int e^{ipx^{'}/\hbar}\Psi^{*}dx^{'} \right ]\left [ \frac{1}{\sqrt{2\pi \hbar}}\int i\hbar\frac{\partial }{\partial p} e^{-ipx/\hbar}\Psi dx \right ]dp        (2)

Consider the integrand function in the second set of brackets. In particular,  i\hbar\frac{\partial }{\partial p} e^{-ipx/\hbar}.

Using the fact that \frac{\mathrm{d} }{\mathrm{d} p}\left ( e^{-ipx/\hbar} \right )= -\frac{ix}{\hbar} e^{-ipx/\hbar} we can rewrite (2) using x e^{-ipx/\hbar} in place of i\hbar \frac{\partial }{\partial p}\left ( e^{-ipx/\hbar} \right ) ...

\int \left [ \frac{1}{\sqrt{2\pi \hbar}}\int e^{ipx^{'}/\hbar}\Psi^{*}dx^{'} \right ]\left [ \frac{1}{\sqrt{2\pi \hbar}}\int x e^{-ipx/\hbar} \Psi dx \right ]dp

Simplifying further and using Fubini's Theorem gives

\frac{1}{2\pi \hbar}\int \int \int \left [ e^{ipx^{'}/\hbar}\Psi^{*} x e^{-ipx/\hbar}\Psi \right ]dx^{'}dxdp     (3)

Letting y = \frac{p}{\hbar} and simplifying the integrand further gives

\frac{1}{2\pi }\int \int \int \left [ e^{-i(x-x^{'})/y} \Psi^{*} x \Psi \right ]dx^{'}dxdy     (4)

Evaluating the y-integral first...

\frac{1}{2\pi }\int e^{-i(x-x^{'})/y}dy = \delta (x-x^{'})

If you're not comfortable with this, check it for yourself by using Plancherel's Theorem to determine the Fourier transform of \delta(x).

Now we have

\int \int \delta(x-x^{'})  \Psi^{*}(x^{'},t) x \Psi (x,t) dx^{'}dx     (5)

But  \int \delta(x-x^{'}) \Psi^{*}(x^{'},t) = \Psi^{*} (x,t)  (see Eq. 3.129 in the first edition and Eq. 3.52 in the second edition), so we have

\int \Psi^{*}x \Psi dx = \int x |\Psi (x,t)|^{2}dx

QED

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