How Does Dirac Notation Impact the Time-Dependent Schrödinger Equation?

[tpg]

I am working through a problem relating to the conservation of probability in a continuity equation. However, I end up with a contradiction when trying to put the following into the Time-Dependent Schrodinger Equation

$$\frac{\partial\psi(x)}{\partial t}=\frac{\partial\left\langle x|\psi\right\rangle }{\partial t}$$

where

$$\left|\psi\right\rangle =\intop_{-\infty}^{\infty}dx\,\psi(x)\left|x\right\rangle$$

If I use the first form, together with

$$i\hbar\frac{\partial\psi(x)}{\partial t} = H\psi(x)$$
and $$-i\hbar\frac{\partial\psi^{*}(x)}{\partial t} = H\psi^{*}(x)$$

I get a sensible nonzero answer, which I believe to be correct.

However, if I start with the second form, I rearrange as follows (I'm pretty sure this step is correct)

$$\frac{\partial\left\langle x|\psi\right\rangle }{\partial t}=\frac{\partial\left\langle x\right|}{\partial t}\left|\psi\right\rangle +\left\langle x\right|\frac{\partial\left|\psi\right\rangle }{\partial t}$$

Then if I use the following form of the TDSE:

$$i\hbar\frac{\partial\left|\psi\right\rangle }{\partial t} = H\left|\psi\right\rangle$$

Which I believe results in

$$\frac{\partial\left\langle x|\psi\right\rangle }{\partial t}=\frac{i}{\hbar}\left(\left\langle x\right|H\left|\psi\right\rangle -\left\langle x\right|H\left|\psi\right\rangle \right)=0$$

This doesn't make any sense to me. Can anyone please explain where I've gone wrong? Many thanks in advance.

 

[Jhero]

Thank you for your post. It seems like you have made a small mistake in your rearrangement of the time-dependent Schrodinger equation. The correct form should be:

\frac{\partial\left\langle x|\psi\right\rangle }{\partial t}=\left\langle x\right|\frac{\partial\left|\psi\right\rangle }{\partial t} + \frac{\partial\left\langle x\right|}{\partial t}\left|\psi\right\rangle

Notice that the order of the terms on the right-hand side has been switched. This is because the derivative is being taken with respect to time, not position. This should fix the contradiction you are seeing.

I hope this helps. Please let me know if you have any further questions or concerns.

[Your Institution]