Electron encountering metal surface (1D Step potential)

[EE18]

Homework Statement

Ballentine Problem 4.3 (which I am self-studying) gives is as follows:

The simplest model for the potential experienced by an electron at the surface of a metal is a step: $W(z) = —V_0 for z < 0$ (inside the metal) and $W(z) =0 for z > 0$ (outside the metal). For an electron that approaches the surface from the interior, with momentum $\hbar k$ in the positive $x$ direction, calculate the probability that it will escape.

Relevant Equations

$$-\frac{h^2}{2M}\frac{d^2\psi}{dx^2} + W\psi = E\psi \implies -\frac{h^2}{2M}\frac{d^2\psi}{dx^2} = (E-W)\psi$$

I am struggling with how to go about this; in particular, I'm not sure I understand what state is being alluded to when Ballentine says "For an electron that approaches the surface from the interior, with momentum $\hbar k$ in the positive $x$ direction, calculate the probability that it will escape." Presumably I am supposed to find some eigenstate of $H$ here, but am I to take a state with $E>|V_0|$ or $E<|V_0|$? I would imagine we're interested in a bound state (so $-V_0<E<0$)?

 

[vela]

You should probably consider both cases. You will find that the probability is 0 if $E<0$ as you may intuitively expect.

 

[EE18]

You should probably consider both cases. You will find that the probability is 0 if $E<\lvert V_0 \rvert$ as you may intuitively expect.

I see; how then is this a model for a surface of a metal when in general the electron states in a metal are bound?

 

[hutchphd]

Can you solve the Shrodinger Equation for a finite step potential at x=0? There are no "bound" (localized) states per se. There are states that fill the solid (for E<0) and states that fill all space (for E>0). For the latter states you can find define the Transmission and Reflection asymptotically for large negative and positive z

 

[EE18]

Can you solve the Shrodinger Equation for a finite step potential at x=0? There are no "bound" (localized) states per se. There are states that fill the solid (for E<0) and states that fill all space (for E>0). For the latter states you can find define the Transmission and Reflection asymptotically for large negative and positive z

I have this so far and will continue on. Does this seem like a reasonable argument for excluding the $E<0$ possibility rigorously?

 

[vela]

I see; how then is this a model for a surface of a metal when in general the electron states in a metal are bound?

If you're assuming the electron is bound, then by assumption it can't escape.

You might imagine a case where the metal is hot enough so that some fraction of the electrons have enough thermal energy to escape if they reach the surface.

 

[EE18]

If you're assuming the electron is bound, then by assumption it can't escape.

You might imagine a case where the metal is hot enough so that some fraction of the electrons have enough thermal energy to escape if they reach the surface.

I see, that makes sense -- thank you! I am so used to seeing artificial ground state textbook cases in solid state physics texts that I didn't think of that.

 

[vela]

I have this so far and will continue on. Does this seem like a reasonable argument for excluding the $E<0$ possibility rigorously?

Your solution for $x<0$ (you mistakenly said $x>0$ again for the second case) is wrong. Also, the only time you're going to get discontinuities in $\psi'$ is when you have some sort of potential involving an infinity, which you don't have here. You want to construct a solution where $\psi$ and $\psi'$ are continuous at $x=0$.

 

[EE18]

Your solution for $x<0$ (you mistakenly said $x>0$ again for the second case) is wrong. Also, the only time you're going to get discontinuities in $\psi'$ is when you have some sort of potential involving an infinity, which you don't have here. You want to construct a solution where $\psi$ and $\psi'$ are continuous at $x=0$.

Sorry for not being clear, you are right re $x>0$. In general, my strategy was to give a solution for $x>0$ and show that it could not be stitched together with the $x<0$ solution in such a way as $\psi'$ was continuous. Is that correct? Also why is my (intended) $x<0$ solution wrong?

 

[hutchphd]

OMG. Why would you redefine the potential halfway through the problem. Please use LateX and start again if you want help here..... There are no bound states. Scattering from a potential step is treated in almost every textbook.

 

[vela]

Sorry for not being clear, you are right re $x>0$. In general, my strategy was to give a solution for $x>0$ and show that it could not be stitched together with the $x<0$ solution in such a way as $\psi'$ was continuous. Is that correct? Also why is my (intended) $x<0$ solution wrong?

Oh, I didn't notice you redefined the potential. With the new potential, you have to have $E>0$. That should make clear why your $x<0$ solution is wrong.

 

[EE18]

Oh, I didn't notice you redefined the potential. With the new potential, you have to have $E>0$. That should make clear why your $x<0$ solution is wrong.

I agree $E>0$; to be clear, I was trying to show in this part of the solution (it's not a complete solution yet) why $E<0$ is impossible since it's not excluded on the basis of e.g. Hermitian operators having real eigenvalues a priori.