jg370
Feb12-11, 09:27 AM
1. The problem statement, all variables and given/known data
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}}{\par tial t}\rangle
Now, the Schrodinger 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\ha t{Q}\psi +\langle\psi\lvert\frac{\partial\hat{Q}}{\partial t}\psi\rangle+\langle\psi\lvert\hat{Q}(\frac{1}{\i math\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\lve rt\hat{Q}\hat{H}\psi\rangle
2. Relevant 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".
3. 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
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
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}}{\par tial t}\rangle
Now, the Schrodinger 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\ha t{Q}\psi +\langle\psi\lvert\frac{\partial\hat{Q}}{\partial t}\psi\rangle+\langle\psi\lvert\hat{Q}(\frac{1}{\i math\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\lve rt\hat{Q}\hat{H}\psi\rangle
2. Relevant 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".
3. 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
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution