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# 1D Schrodinger Equation watch

1. So lately I've been working through a book on quantum mechanics and I have a couple of questions about solving the 1D Schrodinger wave equation.

1. When solving the equation for a rectangular potential barrier:

I understand that
and for the classically forbidden region and by inspection a suitable wavefunction would be and then by differentiating to find the second derivative and substituting, simplifying, solving for k, etc.. you get to a general solution of

but how would you then solve for the probability that the particle has tunnelled through the potential barrier from this?

2. When solving the equation for a quantised harmonic oscillator:

and again by inspection a suitable wavefunction is and I know how to solve for but I don't get why this particular wavefunction leads to the value of that is the zero-point energy.

Would someone please explain these to me? Thanks.
2. There's a few ways of approaching this, but it's quite a bit more Mathematically Heavy than for a rectangular problem.

Step 1: De-dimensionalise

We can change a few things in this problem to make it simpler; in this case we choose that and then we substitute this into the equation; by de-dimensionalising it allows us to compare some quantities later which we wouldn't be able to do otherwise.

Therefore, we get:

We multiply through by

and then choose that
which gives that

And choose that

This simplifies our differential equation significantly; we get that

Step 2: Look at the behaviour as y -> +/- infinity

Outside of the potential, we need the wavefunction to die away, so we assume that and therefore we can say that
which has the solution

This accounts for the behaviour outside of the potential but not that inside; we say that and obviously we know psi. By substituting into the Schrodinger equation, we can get a differential equation in phi;

This is very similar to Hermite's Equation, and you can assume that there exists a power series solution, and get a recursion relationship from it that gives that, for n >= 0,

From which the coefficients can be determined in the wavefunction.
Now comes the part which you asked about; This series terminates when , and we know from what we set earlier that , and therefore

I'm not sure I've explained the last part very well, so ask me any questions if you want!
3. (Original post by doomhalo)
There's a few ways of approaching this, but it's quite a bit more Mathematically Heavy than for a rectangular problem.

Step 1: De-dimensionalise

We can change a few things in this problem to make it simpler; in this case we choose that and then we substitute this into the equation; by de-dimensionalising it allows us to compare some quantities later which we wouldn't be able to do otherwise.

Therefore, we get:

We multiply through by

and then choose that
which gives that

And choose that

This simplifies our differential equation significantly; we get that

Step 2: Look at the behaviour as y -> +/- infinity

Outside of the potential, we need the wavefunction to die away, so we assume that and therefore we can say that
which has the solution

This accounts for the behaviour outside of the potential but not that inside; we say that and obviously we know psi. By substituting into the Schrodinger equation, we can get a differential equation in phi;

This is very similar to Hermite's Equation, and you can assume that there exists a power series solution, and get a recursion relationship from it that gives that, for n >= 0,

From which the coefficients can be determined in the wavefunction.
Now comes the part which you asked about; This series terminates when , and we know from what we set earlier that , and therefore

I'm not sure I've explained the last part very well, so ask me any questions if you want!

I was lacking a lot of the mathematical techniques back then but now I get it. Thanks for the reply.

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