Sunday, June 27, 2010

Some more quantum mechanics!


We saw in a few earlier blogs a basic intro to quantum mechanics. One of the most important equations is the Schrodinger wave equation which is usually written in the abbreviated, operator form:

H_op(U) = E_op(U)

Here H_op is what we call a Hamiltonian or special mathematical operator (combining kinetic and potential energy functions). 'PSI' (U) is the wave function. H_op operates on PSI and yields the quantity shown on the right side. Which is the product of a pure energy operator, E_op by PSI.

From the resulting computations, one can obtain a series of eigenfunctions (U_n) and from these, eigenvalues. When we examine the results- we find an energy expression of the form:

E_n = (Quantity) n^2

Here the bracketed quantity is a constant, the left hand side shows E_n or some energy E referenced to a level (principal quantum number) n. The key point here is the eigenvalues are referenced to energy levels, say in the hydrogen atom.

What this shows is the quantization of atomic energy: energy released by an atom comes in quanta - not continuously. That is, when we observe an atom - say like hydrogen- we should find "lines" of discrete energy emission corresponding to these energies.

In the accompanying diagram I show a number of the eigenfunctions obtained from the Schrodinger wave equation, and the subscripts reference the putative energy levels which can also be described in terms of the quantum numbers (n, l m_l) shown on the left side of the table.

Note that no two discrete levels display the same set of quantum numbers. This also has bearing on one of the most important quantum principles - the Pauli Exclusion Principle. It basically states that: In a multi-electron atom no two electrons can have the same quantum state.

Now, since quantum state is specified by the quantum number set (n, l, m_l) that means no two electrons can have the same set.

What are some of the implications of applying the principle?

One of the more interesting is that if one writes out the wave function for any number of particles, N, we can also write it in terms of separated wavefunctions, e.g.

U(1, 2, 3,....N) = U(1) U(2) U(3)......U(N)

Now, suppose we have some 2-electron state with one electron in state (a) and the other in state (b), such that:

U(1a, 2b)

Then, the probability density is the same whether we we say "electron 1 is electron 2", or "electron 2 is electron 1". Thus:

[U(1,2)]^2 = [U(2,1)]^2

we have here: U(2,1) = U(1,2) as a symmetric wave function and

U(2,1) = - U(1,2) as an asymmetric wave function.

Now, look at the two electrons relative to states I, and II:

U_I = U_a(1)U_b(2)

U_II = U_a(2) U_b(1)

compare now the full solution for symmetric linear combination:

Symmetric:

U_s = 1/(2)^1/2 [U_a(1)U_b(2) + U_a(2) U_b(1)]

which for the above means a = b

then, antisymmetric:

U_AS = 1/(2)^1/2 [U_a(1)U_b(2) - U_a(2) U_b(1)]

for which we find a is never identically equal to b.

Thus, it must follow all electron wavefunctions must be anti-symmetric in order to satisfy the Pauli principle, and further all the electrons must have half-integral (e.g. n/2) spins.

Without the consequences of the Pauli Exclusion principle, no chemistry would be possible - since this is the principle which underlies why chemical elements differ!