note08_1 - Introduction to Quantum and Statistical...

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Introduction to Quantum and Statistical Mechanics Handout 8 Approximation Methods in Quantum Mechanics Assigned Reading: Chaps. 13 – 16, pp. 247 – 304. 8.1 Overview We have learned the basic principles in quantum mechanics and how to analytically solve the Schrödinger equation in simple potentials. The general steps include finding the eigenfunctions and eigenvalues of the time-independent Schrödinger equation according to the boundary condition (boundary value problem), express the initial condition as the expansion of the eigenfunctions, and then the solution of the wave functions can be obtained by adding ( 29 / exp t iE n - to the eigenfunction expansion (initial value problem). The expectation value of all measurable quantities can then be calculated from the wavefunction. However, the Schrödinger equation can become unsolvable analytically either due to the increase of the number of variables (you have seen the multiple-particle effect in phonons or you can imagine a pack of electrons and ions forming plasma in 3D) or the complexity of the potential. In fact, there are only a handful of potentials that we have reasonably tractable solutions. For most practical problems, a good and accurate “guess” or “approximation” is not often necessary, but also gives the best intuition when we try to design a quantum system. Complex potential can often be treated with the perturbation method. If the system Hamiltonian can be written as: 1 0 ˆ ˆ ˆ H H H + = (8.1) where 0 ˆ H has known eigenfunctions and eigenvalues: ) ( ) ( ˆ 0 x E x H n n φ = (8.2) 1 ˆ H is called the perturbation Hamiltonian. For the wave function ψ (x) that satisfies the system Hamiltonian, i.e., ) ( ) ( x E x H = , since n (x) will form a complete set, (x) can be expressed as the linear combination of n (x) as long as 1 ˆ H does not change the continuity at boundaries and behavior at infinity: = = N n n n x c x 1 ) ( ) ( (8.3) However, since we do not know the form of (x), we cannot use the braket < k | method to find c n . Substituting this expansion into the Schrödinger equation of H (x)=E (x) : 1
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( 29 = + n n n n n n c E c H H φ 1 0 ˆ ˆ ( 29 = - n n n n n n n H c c E E 1 ˆ (8.4) Now we can use the bracket < k | method to simplify the summation, i.e., we will multiply both sides with k * and integrate over the entire x space, ( 29 = - n n k n k k H c c E E | ˆ | 1 (8.5) Notice that we know the form of k and n and hence the braket can be evaluated. For each index k we will obtain an equation containing the N+1 unknowns of c n and E . What we obtain here will be N linear equations plus the normalization to find c n and E . This is why we call n k H | ˆ | 1 as the matrix element.
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note08_1 - Introduction to Quantum and Statistical...

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