How to Derive the Continuity Equation for a Particle in a 1D Potential?

[whatisreality]

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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger equation in order to answer this question? If I do... how do I do it??

 

[Fightfish]

Then I thought I would solve the Schrödinger equation.

Why do you think solving the Schrödinger equation would be useful? Besides, it is not possible to solve the Schrödinger 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?

 

[whatisreality]

Why do you think solving the Schrödinger equation would be useful? Besides, it is not possible to solve the Schrödinger 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?

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 Schrödinger equation, the potential was still in the equation at the end. I'll post where I got to...

 

[whatisreality]

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

 

[whatisreality]

Why do you think solving the Schrödinger equation would be useful? Besides, it is not possible to solve the Schrödinger 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?

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 Schrödinger equation, so I can't find $\rho$.

 

[Fightfish]

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

 

[whatisreality]

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

Got it, thank you!