Stationary State Approximate Methods Chapter 2 Lecture 2.1 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

Approximate Methods: Approximate methods for stationary states corresponds to time independent Hamiltonian. Examples: perturbation theory, variational method, and the WKB method. Perturbation theory: Exactly sovable part + small correction If we are not able to rediuce the hamiltonian of the Problem to exactly solable part+ small corection then We use variation method or WKB method

The variational method Useful in estimating the energy eigenvalues of the ground state and the first few excited states of a system for which one has only a qualitative idea about the form of the wave function. The WKB method is useful for finding the energy Eigenvalues and wave functions of systems for which the classical limit is valid. Unlike perturbation theory, the variational and WKB methods do not require the existence of a closely related Hamiltonian that can be solved exactly.

Variational Method (Rayleigh–Ritz method): The variational method is useful for determining upper bound values for the eigen energies of a system whose Hamiltonian is known whereas its eigen values and eigenstates are not known. It is particularly useful for determining the ground state. It becomes quite cumbersome to determine the energy levels of the excited states.

We will not directly solve Schrodinger Eq ---------(1) We use variational scheme and consder variation Eq -----(2) expectation value of energy -------(3)

If depend on parameter α then E(ψ) too If depend on parameter α then E(ψ) too. We vary α so that E(ψ) get minimize. Minimum value of E will give approximation to the upper Limit for true value of energy. For any trial function the energy given by (3) is Always larger then then the ground state energy E 0 . -----(4) We will prove (4) now.

Expanding in terms of eigenstates of hamiltonian H, We have -----------(5) ----------(6) For non-degenerate one-dimensional bound state E0<=En . Thus we can write -------(7) Which is Eq. (4).

Steps is using variational method: Choose appropriate trial function -----------(8) (2) Calculate energy using (4) This wil depend on parameters ------(9) .

(3) Minimize E obtained in step 2 by varying parameters. --------(10) (4) Use in (9) to obtain approximate Energy. This give an upper bound on exact energy E0 Ground state eigenfunction is

Harmonic oscillator: Ground state energy of hermonic oscillator using the variational method. Ans: Trial function ------(1) A is normalization constant. Ground state energy E0(α) is given by ---------(2)

--------(3) Minimize E(α) w.r.t. α ------(4) -------(5)

Using (5) in (1) and (3), -----------(6)

Important point: When first derivative of Wave function is discontinuous, we should not use the formula --(1) to find kinetic energy. We should use --(2)

Eq. (1) and (2) are identical ---(3) In three dimension for we use Gauss theorem ----------(4)

We write ----(5) For , surface integral vanishes. Thus, ----(6)

Example: Find an upper on the ground state energy of an infinite square will potential With triangular wave function as trail ----(1)

Normalization constant --------(2) First derivative (Note the discontinuity) ----(3)

We can write Eq (3) in form of step functions ---(4) Differentiating above and using We get -------(5)

Thus expectation value of Hamiltonian ---(6) Which gives estimated upper bound on Ground state energy using variation method. Recall, true ground state energy is ----(7)

Corollary (How to use variational method To find the energy of excited states): (a)

Exercise:

Hydrogen atom: Consider trial function ---(1) Here α is some parameter. Expectation value of energy ----(2)

Now P.E. ---(3) ---(4) Kinetic energy ----(5) Where ---(6)

From (5) ---(7) Using (4) and (7) in (2) ----(8) Minimizing ---(9) We have ----(10)

Thus ---(11)