Electron encountering metal surface (1D Step potential)

In summary: 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...an.
  • #1
EE18
112
13
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##)?
 
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  • #2
You should probably consider both cases. You will find that the probability is 0 if ##E<0## as you may intuitively expect.
 
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  • #3
vela said:
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?
 
  • #4
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
 
  • #5
hutchphd said:
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?

Screen Shot 2023-03-27 at 2.16.27 PM.png
 
  • #6
EE18 said:
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.
 
  • #7
vela said:
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.
 
  • #8
EE18 said:
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##.
 
  • #9
vela said:
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?
 
  • #10
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.
 
  • #11
EE18 said:
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.
 
  • #12
vela said:
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.
 

1. What is an electron encountering a metal surface (1D Step potential)?

An electron encountering a metal surface (1D Step potential) refers to the process of an electron moving from a region of lower energy to a region of higher energy when it encounters a metal surface. This can occur when an electron is incident on a metal surface with a step-like potential barrier.

2. What is the significance of studying electron encountering metal surfaces?

The study of electron encountering metal surfaces is important in understanding the behavior of electrons in materials, particularly in the field of condensed matter physics. It can also provide insights into the properties of metals and their applications in various technologies.

3. How does the energy of the electron change when it encounters a metal surface?

When an electron encounters a metal surface, its energy changes depending on the potential barrier of the metal surface. If the potential barrier is higher than the energy of the electron, the electron will be reflected back. If the potential barrier is lower, the electron will be transmitted through the surface.

4. What factors affect the behavior of an electron encountering a metal surface?

The behavior of an electron encountering a metal surface is affected by several factors such as the energy of the electron, the potential barrier of the metal surface, and the properties of the metal itself, such as its atomic structure and electronic band structure.

5. How is the phenomenon of electron encountering metal surfaces used in practical applications?

The phenomenon of electron encountering metal surfaces is used in various practical applications, such as in electronic devices like transistors and solar cells, as well as in materials characterization techniques like scanning tunneling microscopy. It is also crucial in the development of new materials and technologies in fields such as nanotechnology and quantum computing.

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