Schrodinger equation in position representation

[MatinSAR]

Homework Statement

Derive schrodinger equation in position representation.

Relevant Equations

Schrodinger equation

Hello.

General form of the equaion is $i \hbar \dfrac {\partial}{\partial t} | \psi (t) \rangle = \hat H | \psi (t) \rangle$
According to my book $\hat H$ in position space is $- \dfrac {\hbar ^2}{2m} \nabla^2 + V( \hat {\vec R} , t)$ so potensial V is a function of operator $\hat R$ and time $t$. But in another page it used $\hat V(\vec r , t )$ instead, and I cannot understand why.

 

[anuttarasammyak]

What are R and r in the text ?

 

[MatinSAR]

What are R and r in the text ?

$\hat{R}$ is a linear Hermitian operator corresponding to the observable $\vec r$.

 

[anuttarasammyak]

Thanks. In position presentation, applying operator R is multiplying its eigenvalue parameter r. e.g. a harmonic oscillator
\hat{V}=V(\hat{\mathbf{R}})=\frac{k}{2}\hat{\mathbf{R}}^2=\frac{k}{2}\mathbf{r}^2=V(\mathbf{r})=\hat{V}(\mathbf{r})

 

[MatinSAR]

Thanks. In position presentation, applying operator R is multiplying its eigenvalue parameter r. e.g. a harmonic oscillator
\hat{V}=V(\hat{\mathbf{R}})=\frac{k}{2}\hat{\mathbf{R}}^2=\frac{k}{2}\mathbf{r}^2=V(\mathbf{r})

Are you saying that $\hat V(\mathbf r)$ is same as $V( \hat R)$?

 

[anuttarasammyak]

Yes, in position representation.

 

[MatinSAR]

Yes, in position representation.

I've just seen your edit in post #4. It's clear. Thank you for your time.