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I've been mulling over this for a while and getting nowhere.

Here is the question: an electron is bound in a square well of width $1.95\:\mathrm{nm}$ and depth $U = 6E_∞$. If the electron is initially in the ground level and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the well?

In order to find the wavelength I need the energy difference between $U$ and $E_1$. The thing that confuses me the most is that according to the solutions, $E_1 = 0.625E_∞$. How do you find this?

Using the formula $E = n^2h^2/8mL^2$ I have $E_1 = 1.58 \times 10^{-20}\:\mathrm{J}$. What am I missing?

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What you are missing is that this is not an infinite square well.

The formula you are using is the formula for the energy states in an infinite square well potential, where the wave function of the electron is entirely contained within the well, and the constraint is that the wave function is zero at the edges of the well.

But the problem posed is very much a finite potential well of potential depth $U$. In this scenario, which means you need to find expressions for the wave function both inside and outside the square well (rather than just inside, for the infinite square well) and to have them connect smoothly (same value of the wave function at $x=\pm\frac{width}{2}$).

Any introductory textbook on quantum mechanics will almost certainly have a section on the finite potential well. Or you can consult the Wikipedia page on the finite potential well which also covers the derivation.

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  • $\begingroup$ Oh! That makes a lot more sense. For some reason the fact that U is multiplied by E at infinity made me think of infinite well. Unfortunately my textbook University Physics doesn't go into much detail and just gives the numbers for finite well energies without explanation... so I'll do a bit of digging. Thanks. $\endgroup$ – jebbbs Oct 31 '15 at 9:24

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