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The Schrödinger equation and its solutions for the hydrogen atom can be written as
The subscripts are so-called quantum numbers. Because the subscripts n, l, and m can take on more than one value we have many solutions. It can be shown that n is always a positive integer; i.e., n = 1,2,3,.... It's called the principal quantum number . The quantum number l is always a non-negative integer. It's called the angular quantum number . For a given n it is restricted to l = 0,1,2,...,n-1. The number m is also always an integer, but its value range from -l to l; i.e., m = -l,-l+1,-l+2,...,l-2,l-1,l. It is called the magnetic quantum number . Combining n, l, and m gives the wave functions Y1,0,0, Y2,0,0, Y2,1,-1, Y2,1,0, Y2,1,1, Y3,0,0, Y3,1,-1, Y3,1,0, Y3,1,1, Y3,2,-2, Y3,2,-1, Y3,2,0, Y3,2,1, Y3,2,2, etc. The functions are grouped in shells and subshells. Wave functions with the same n belong to the same shell. Wave functions with the same n and l belong to the same subshell. The subshells are called 1s, 2s, 2p, 3s, 3p, 3d, etc. The number is the principal quantum number n. The letter indicates the angular quantum number l.
The energy En that is a solution of the Schrödinger equation is the same energy as found in the Bohr model of the hydrogen atom; i.e.,
The energy En of the wave function Ynlm depends only on n. This means that all wave function that belong to the same shell have the same energy. If two or more wave function have the same energy one says that they are degenerate . The degeneracy of a level is the number of wave functions with the same energy. The degeneracy can easily be determined for the hydrogen atom. We only have to count for a given n how many different combination we can have of the l and m quantum numbers.
The simplest case is n = 1. Because l is a non-negative integer smaller than n, the only possibility is l = 0. From this we immediately get m = 0 as well, because m is an integer in the range [-l,l]. So the only combination with n = 1 is (n,l,m) = (1,0,0). (We had already seen this; Y1,0,0 or 1s was the only wave function we had found with n = 1.) This means that the degeneracy for the level with n = 1 is 1, and the energy is -me4/8e02h2.
For n = 2 we can have l = 0, but also l = 1. If l = 0 we must have m = 0, but if l = 1 then m = -1, 0, or 1. This means that we have four wave functions with n = 2. So the degeneracy for the level with n = 2 is 4, and the energy is -me4/32e02h2. For n = 3 we can have l = 0, 1, or 2. For l = 0 we have one, and for l = 1 we have three possibilities for m. In general we have 2l+1 possibilities. This means that for l = 2 we have five possible values for m. So for n = 3 we have a degeneracy of 1+3+5 = 9, and an energy of -me4/72e02h2.
People with a feeling for patterns may already suspect the degeneracy for n = 4. For n = 1, 2, and 3 we found 1 ( = 12), 4 ( = 22), and 9 ( = 32). Indeed the degeneracy for n = 4 is 16. It is not hard to prove that the degeneracy is always equal to n2. For a given n we need to sum the number of possible values for m for l = 0,1,2,...,n-1. The number of values is 2l+1 for a given l. So the degeneracy is
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l = 0
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The energies of the levels get ever closer to 0 for increasing values of n, and also get closer to each other.
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