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I have the propagator for the harmonic oscillator.

$$K(x_f,x_0,t)=\sqrt{\frac{m\omega}{2 \pi \hbar \sin{wt}}}\exp\left(\frac{i}{\hbar}\frac{m\omega}{2 \sin{\omega t}}((x_0^2+x_f^2)\cos\omega t-2x_0x_f)-\frac{i\pi}{4} \right)$$

and

$$K(x_f,x_0,t)=<x_f |e^{-\frac{i}{\hbar}\hat{H}t} |x_0 > $$

and the Schrodinger equation

$$\hat{H}\psi(\textbf{q},t)=i\hbar \frac{\partial}{\partial t}\psi(\textbf{q},t) $$

This may seem silly but I just cannot see from my notes how you can the propagator satisfies this Schrodinger equation. I think it may come from some lack of understanding as to what the propagator means.

Trajan
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  • How did you derive the propagator? – Ryan Unger Feb 04 '15 at 23:43
  • Using the action of the Lagrangian, and a couple of clever tricks. This is stuff from lecture notes – Trajan Feb 04 '15 at 23:48
  • For a general proof that path integrals reproduce the Schroedinger equation, see http://physics.stackexchange.com/q/163075/ – Ryan Unger Feb 04 '15 at 23:52
  • What you have to do is put your propagator where I had my general one. After probably pages and pages of math, you can verify it. I don't know of a more elegant method. – Ryan Unger Feb 05 '15 at 00:00
  • Voting to close as unclear what you're asking because nobody ever said a propagator should satisfy the Schrödinger equation. – ACuriousMind Feb 05 '15 at 00:19
  • Read about greens functions. – lionelbrits Feb 05 '15 at 00:24
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    @ACuriousMind: I think he means "satisfy" in the sense that the convolution of the propagator with an initial solution yields another solution. – Ryan Unger Feb 05 '15 at 00:50
  • Related: http://physics.stackexchange.com/q/65489/2451 and links therein. – Qmechanic Feb 05 '15 at 01:25
  • I was looking more for an explicit way of evaluating the propagator on the Schrodinger equation than a more general explanation. Really I think I was just looking for a hint/first step of how I can "put" the propagator as shown in the question into the Schrodinger equation. I have a gut feeling that this should be a "simple" substitution. – Trajan Feb 08 '15 at 14:41
  • @ACuriousMind Could you explain why "a propagator should (not) satisfy the SE"? This seems strange as it is pretty much the question I asked. I believe this is the problem that is stopping me from solving this question as I cannot see how to do this. I dont expect full solutions, nor is this work for credit. – Trajan Feb 08 '15 at 15:06
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    The propagator doesn't follow the Schroedinger equation, it obeys $\psi(x,t)=\int\psi(x',t')K(x,t;x't')dx'$. To check your propagator, you need to solve that integral. – Kyle Kanos Feb 08 '15 at 15:29
  • @KyleKanos Thank you that was the explanation I was looking for – Trajan Feb 08 '15 at 15:39

1 Answers1

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The propagator does satisfy the Schrodinger equation for most values of x and t...

The easiest way to show this is to let $\hbar=m=\omega=1$. Also, we can work with $\sqrt{2\pi}e^{i\pi/4}K\to K$ instead of $K$ to clean up the mess a little further. Further, let $x_f\to x$ and $x_0 \to 0$.

Then let: $K=fe^{ig}$ with

$f=\frac{1}{\sqrt{\sin(t)}}$

and

$g=\frac{x^2\cos(t)}{2\sin{t}}$

And, since $m=\omega=1$, we want to show that:

$i\dot K = -\frac{1}{2}K''+\frac{x^2}{2}K$

Okay...

$\dot K=\frac{K}{2}\left({\frac{-\cos(t)}{\sin(t)}-i\frac{x^2}{\sin^2(t)}}\right)$

And...

$K''=K\left({i\frac{\cos}{\sin}-x^2\frac{\cos^2}{\sin^2}}\right)$

And... I'll leave the rest up to you. But, I assure you, you will find that: $HK=i\dot K$

...of course, this is not actually, true since actually: $HK=i\dot K+i\delta(x_f-x_0)\delta(t)$

And... I'll leave it up to you to reconcile this issue...

hft
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  • Sorry its not clear "for most values". I was expecting this to be some kind of "simple" substitution. Your step $\hbar=m=\omega=1$ is very useful. – Trajan Feb 08 '15 at 14:40
  • Ah, I said "for most values" because K only really satisfies the schrodinger equation if the delta functions (see the very last equation) are zero. For example, you have problems at "t=0". – hft Feb 09 '15 at 02:15