Question related to Schrodinger equation

[davon806]

Homework Statement

Homework Equations

The Attempt at a Solution

It is a short question so I don't suppose it is difficult. However, I don't really understand what it is asking for :
1.The TDSE itself is already a 2nd-order differential equation (if you substitute the terms back into H).
2. Usually, the equation in the first line should be ψ( r, t) instead of ψ(t). So the question starts from a time-dependent function then proceed to a general case, in which the spatial part also comes into play?

I thought the question is asking you how to derive the TISE Hψ = Eψ. Should I just take the scalar product
< r | ψ(t)> then proceed to the regular steps like separation of variables? Or there is another way to interpret this question?

Thanks

 

[Ssnow]

Hi, first the hamiltonian in dirac equation is $\hat{H}=\hat{H}(t)$. You can start to write $|\psi(t)\rangle=\int dt \psi(r,t)|r\rangle$ with $\langle r'|r\rangle=\delta(r'-r)$ and $\hat{p}|\psi(t)\rangle=-i\hbar \int dr\frac{\partial}{\partial r} \psi(r,t) |r\rangle$. So you can proceed to write $\hat{H}$ for a particle in one dimension $\frac{\hat{p}^2}{2m}+V(r,t)$ substituting the previous expressions...

details are in

http://wiki.physics.fsu.edu/wiki/index.php/The_Schrödinger_Equation_in_Dirac_Notation

Ssnow