How to Find Square-Well Energy Levels

In quantum physics, if you know the boundary conditions of a square well, you can find theenergy levels of an electron.

The equation

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tells you that you have to use the boundary conditions to find the constants A and B. What are the boundary conditions? The wave function must disappear at the boundaries of an infinite square well, so

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The fact that

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tells you right away that B must be zero because cos(0) = 1. And the fact that

image3.png

tells you that

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Because sine is zero when its argument is a multiple of

image5.png

this means that

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Note that although n = 0 is technically a solution, it yields

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for all x, which is not normalizable, so it’s not a physical solution — the physical solutions begin with n = 1.

This equation can also be written as

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And because

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you have the following equation, where n = 1, 2, 3, ... — those are the allowed energy states. These are quantized states, corresponding to the quantum numbers 1, 2, 3, and so on:

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Note that the first physical state corresponds to n = 1, which gives you this next equation:

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This is the lowest physical state that the particles can occupy. Just for kicks, put some numbers into this, assuming that you have an electron, mass

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confined to an infinite square well of width of the order of the Bohr radius (the average radius of an electron’s orbit in a hydrogen atom); let’s say

image13.png

gives you this energy for the ground state:

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That’s a very small amount, about 4 electron volts (eV — the amount of energy one electron gains falling through 1 volt). Even so, it’s already on the order of the energy of the ground state of an electron in the ground state of a hydrogen atom (13.6 eV), so you can say you’re certainly in the right quantum physics ballpark now.

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