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whatisreality

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## Homework Statement

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}##.

## Homework 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*.

## 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??

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