chapter5 - Chapter 5 APPROXIMATION METHODS FOR STATIONARY...

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Chapter 5 APPROXIMATION METHODS FOR STATIONARY STATES As we have seen, the task of prediciting the evolution of an isolated quantum mechan- ical can be reduced to the solution of an appropriate eigenvalue equation involving the Hamiltonian of the system. Unfortunately, only a small number of quantum mechanical systems are amenable to an exact solution. Moreover, even when an exact solution to the eigenvalue problem is available, it is often useful to understand the behavior of the system in the presence of weak external f elds that my be imposed in order to probe the structure of its stationary states. In these situations an approximate method is required for calculating the eigenstates of the Hamiltonian in the presence of a perturbation that renders an exact solution untenable. There are two general approaches commonly taken to solve problems of this sort. The f rst, referred to as the variational method, is most useful in obtaining information about the ground state of the system, while the second, generally referred to as time-independent perturbation theory, is applicable to any set of discrete levels and is not necessarily restricted to the solution of the energy eigenvalue problem, but can be applied to any observable with a discrete spectrum. 5.1 The Variational Method Let H be a time-independent observable (e.g., the Hamiltonian) for a physical system having (for convenience) a discrete spectrum. The normalized eignestates {| φ n i} of H each satisfy the eigenvalue equation H | φ n i = E n | φ n i (5.1) where for convenience in what follows we assume that the eigenvalues and corresponding eigenstates have been ordered, so that E 0 E 1 E 2 ··· . (5.2) Under these circumstances, if | ψ i is an arbitrary normalized state of the system it is straightforward to prove the following simple form of the variational theorem :t h e mean value of H with respect to an arbitrary normalized state | ψ i is necessarily greater than the actual ground state energy (i.e., lowest eigenvalue) of H , i.e., h H i ψ = h ψ | H | ψ i E 0 . (5.3) The proof follows almost trivially upon using the expansion H = X n | φ n i E n h φ n | (5.4) of H in its own eigenstates to express the mean value of interest in the form h H i ψ = X n h ψ | φ n i E n h φ n | ψ i = X n | ψ n | 2 E n , (5.5)
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134 Approximation Methods for Stationary States and then noting that each term in the sum is itself bounded, i.e., | ψ n | 2 E n | ψ n | 2 E 0 , so that X n | ψ n | 2 E n X n | ψ n | 2 E 0 = E 0 (5.6) where we have used the assumed normalization h ψ | ψ i = P n | ψ n | 2 =1 of the otherwise arbitrary state | ψ i . Note that the equality holds only if | ψ i is actually proportional to the ground state of H . Thus, the variational theorem proved above states that the ground state minimizes the mean value of H taken with respect to the normalized states of the space. This has interesting implications. It means, for example, that one could simply choose random vectors in the state space of the system and evaluate the mean value of H with respect to each. The smallest value obtained then gives an upper bound for the ground state
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chapter5 - Chapter 5 APPROXIMATION METHODS FOR STATIONARY...

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