Solving Dirac's Notation Homework: Energy-Time Uncertainty Principle

[jg370]

Homework Statement

Hi,

My textbook provides me with the steps to derive the Energy-Time Uncertainty Principle; while I can follow most of it, I have problem with one particular step.

$$\frac{d}{dt}\langle{Q}\rangle =\frac{d}{dt}\langle\psi\lvert\hat{Q}\psi\rangle$$

$$\frac{d}{dt}\langle{Q}\rangle =\langle\frac{\partial \psi}{\partial t}\lvert\hat{Q}\psi\rangle +\langle\psi\lvert\frac{\partial\hat{Q}}{\partial t}\psi\rangle +\langle\psi\lvert\hat{Q}\frac{\partial\psi}}{\partial t}\rangle$$

Now, the Schrödinger equation says:

$$\imath\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi$$

So, I deduced:

$$\frac{\partial\psi}{\partial t} = \frac{1}{\imath\hbar}*\hat{H}\psi$$

Substituting this in the main equation, we have:

$$\frac{d}{dt}\langle{Q}\rangle = \langle(\frac{1}{\imath\hbar})\hat{H}\psi\lvert\hat{Q}\psi +\langle\psi\lvert\frac{\partial\hat{Q}}{\partial t}\psi\rangle+\langle\psi\lvert\hat{Q}(\frac{1}{\imath\hbar})\hat{H}\psi\rangle$$

$$\frac{d}{dt}\langle{Q}\rangle =- \frac{1}{\imath\hbar}\langle\hat{H}\psi\lvert\hat{Q}\psi +\langle\psi\lvert\frac{\partial\hat{Q}}{\partial t}\psi\rangle+\frac{1}{\imath\hbar}\langle\psi\lvert\hat{Q}\hat{H}\psi\rangle$$

Homework Equations

In the last equation, I have factored $$\frac{1}{\imath\hbar}$$ from the "bra" (first term} and from the "ket" (last term) of above equation and assumed that the sign would be respectively negative and positive? I assumed so because the "bra" is the conjugate of the "ket".

The Attempt at a Solution

Have I assumed corretly? I really so not see any oher possibility. I thank you for your kind assistance

jg370

 

[dextercioby]

What you did is ok. Indeed the scalar picks up a minus and a plus and that's how you end up with the commutator between H and Q.