# Particle in a 1D potential V(x)

1. Jan 13, 2016

### whatisreality

1. The problem statement, all variables and given/known data
There's a particle moving in a 1D potential V(x) with mass m. The particle's normalised wavefunction is ψ(x,t). Use the time dependent Schrodinger equation to show that $\frac{\partial{\rho}}{\partial{t}} + \frac{\partial{j}}{\partial{x}} = 0$
Where
$j(x,t) = -\frac{i\hbar}{2m}(\psi^{*} \frac{\partial{\psi}}{\partial{x}} - \psi \frac{\partial{\psi}}{\partial{x}})$

I also have to show that j(x,t) is real. All I know about j is that it has to be equal to the magnitude of ψ(x,t)$^{2}$.

2. Relevant equations
Time dependent Schrodinger equation:
$i\hbar \frac{\partial{\psi}}{\partial{t}} = -\frac{\hbar^{2}}{2m} \frac{\partial^{2}{\psi}}{\partial{x^2}} + V(x) \psi$

The probability density ρ = |ψ(x,t)|^2 or ψψ*.

3. The attempt at a solution
I'm having a bit of trouble with the calculus element. Pretty sure I'm differentiating wrong AND integrating wrong, but anyway, here's what I got:
First I thought I'd calculate $\frac{\partial{j}}{\partial{x}}$ since j is given. I got
$-\frac{i\hbar}{2m}(\psi^{*}\frac{\partial^{2}{\psi}}{\partial{x^2}}+\frac{\partial{\psi}}{\partial{x^2}}\psi^{*}-\psi\frac{\partial^{2}{\psi^{*}}}{\partial{x^2}}+\frac{\partial{\psi}}{\partial{x}}\frac{\partial{\psi}}{\partial{x}})$.

Then I thought I would solve the Schrodinger equation. But (and I know this isn't exactly a good reason) the question is only worth five marks! Which makes me think that I might not have to solve the Schrodinger equation. And I wouldn't actually know how to solve it anyway...

So is my first calculation right? And do I need to solve the Schrodinger equation in order to answer this question? If I do... how do I do it??

Last edited: Jan 13, 2016
2. Jan 13, 2016

### Fightfish

Why do you think solving the Schrodinger equation would be useful? Besides, it is not possible to solve the Schrodinger equation in closed form for general $V(x)$

There is a mistake in your expression for the current density. There should be a complex conjugation on the second term: $$j(x,t) \equiv -\frac{i\hbar}{2m} \left(\psi^{*} \partial_{x} \psi - \psi \partial_{x} \psi^{*} \right)$$
Calculating $\partial_{x} j(x,t)$ is a good place to start - do correct the complex conjugation mistakes though.
Have you tried calculating $\partial_{t} \rho$?

3. Jan 13, 2016

### whatisreality

So it represents current density! Interesting. OK, I'll sort out the conjugation mistakes - if I add the *, should it be correctly differentiated though?

I have tried calculating ρ for a start, but pretty unsuccessfully. I think I broke some rules along the way, while trying to solve the schrodinger equation, the potential was still in the equation at the end. I'll post where I got to...

4. Jan 13, 2016

### whatisreality

So corrected version for partial derivative of j: $-\frac{i\hbar}{2m}(\psi^{*}\frac{\partial^{2}{\psi}}{\partial{x^2}}+\frac{\partial{\psi^{*}}}{\partial{x}}\frac{\partial{\psi}}{\partial{x}}-\psi\frac{\partial^{2}{\psi^{*}}}{\partial{x^2}}-\frac{\partial{\psi}}{\partial{x}}\frac{\partial{\psi^{*}}}{\partial{x}})$

5. Jan 13, 2016

### whatisreality

As for $\rho$, actually I know my method was wrong. I made ∂Ψ/∂t the subject and then integrated. I don't know how to solve the Schrodinger equation, so I can't find $\rho$.

6. Jan 13, 2016

### Fightfish

You don't need to solve the Schrodinger equation to "find" $\rho$.
$$\partial_{t} (\psi^{*}\psi) = \psi \partial_{t} \psi^{*} + \psi^{*} \partial_{t} \psi$$
Now, use the Schrodinger equation to replace $\partial_{t} \psi^{*}$ and $\partial_{t} \psi$

7. Jan 15, 2016

### whatisreality

Got it, thank you!