approximateMethods

approximateMethods - Physics 125c Course Notes Approximate...

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Physics 125c Course Notes Approximate Methods 040415 F. Porter Contents 1 Introduction 2 2 Variational Method 2 2.1 Bound on Ground State Energy . . . . . . . . . . . . . . . . . 2 2.2 Example: Helium Atom . . . . . . . . . . . . . . . . . . . . . 3 2.3 Other Applications of Variational Method . . . . . . . . . . . 6 2 . 4 V a r i a t i on a lTh e o r em . ...................... 7 2.5 The “Ritz Variational Method” . . . . . . . . . . . . . . . . . 8 3 The WKB Approximation 10 3.1 Example: InFnite Square Well . . . . . . . . . . . . . . . . . . 12 3.2 Example: Harmonic Oscillator . . . . . . . . . . . . . . . . . . 13 4 Method of Stationary Phase 14 4.1 Application: Asymptotic Bessel ±unction . . . . . . . . . . . . 15 4.2 Application: Quantum Mechanics of ±ree Particle Asymptotic W a v un c t i on . ......................... 1 6 4.3 Application: A Scattering Problem . . . . . . . . . . . . . . . 18 5 Stationary State Perturbation Theory 19 5 . 1 N o rm a l i z a t i on. .......................... 2 4 6 Degenerate State Perturbation Theory 25 7 Time-dependent Perturbation Theory 26 7.1 The Time-Ordered Product . . . . . . . . . . . . . . . . . . . 28 7.2 Transition Probability, ±ermi’s Golden Rule . . . . . . . . . . 29 7 . 3 C ou l ombS c a t t e r in g........................ 3 6 7 . 4 D e c a y s............................... 3 7 7.5 Adiabatically Increasing Potential . . . . . . . . . . . . . . . . 39 8 Eigenvalues – Comparison Theorems 41 1
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9 Exercises 45 1 Introduction Typically, problems in quantum mechanics are difficult to solve exactly with analytic methods. We thus resort to approximate methods, or to numerical methods. In this note, I review several approximate approaches. 2 Variational Method There are many applications of the technique of varying quantities to Fnd a useful extremum. This is the gist of the “variational method”. As a means of Fnding approximate solutions to the Schr¨ odinger equation, a common approach is to guess an approximate form for a solution, parameterized in some way. The parameters are varied until an extremum is found. We illustrate this approach with examples. 2.1 Bound on Ground State Energy Given a system with Hamiltonian H , and ground state energy E 0 ,wemay note that for any state vector | ψ ± we must have: ² ψ | H | ψ ± ² ψ | ψ ± E 0 . (1) This suggests that we may be able to use some set of functions ψ , parame- terized in some way, to obtain an upper limit on the ground state energy, even if we cannot solve the problem exactly. With careful choice of “trial” function, we may even be able to get a good approximation to the energy level. Thus, the program is to Fnd the minimum of the quantity in Eqn. 1 over variations in the parameter space to get a “least” upper bound on E 0 for our trial wave functions: δ ± ψ { θ } | H | ψ { θ } ² ± ψ { θ } | ψ { θ } ² δ { θ } =0 . (2) Here, the parameter set to be varied is denoted { θ } . 2
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2.2 Example: Helium Atom The Coulombic Hamiltonian for the helium atom is: H = p 2 1 2 m + p 2 2 2 m - 2 α | x 1 | - 2 α | x 2 | + α | x 1 - x 2 | (3) = - 1 2 m ( 2 1 + 2 2 ) - 2 α | x 1 | - 2 α | x 2 | + α | x 1 - x 2 | , (4) where α = e 2 , m is the mass of the electron, and we are neglecting the motion of the nucleus. Let r 1 = | x 1 | , (5) r 2 = | x 2 | , (6) r 12 = | x 1 - x 2 | . (7) Toward guessing a “good” trial wave function, note that, if the interaction term α/r 12 were not present, the ground state wave function would be simply a product of two hydrogenic ground state wave functions in x 1 and x 2 : ψ ( x 1 , x 2 )= Z 3 πa 3 0 e - Z a 0 ( r 1 + r 2 ) ,a 0 = 1 . (8) There is no reason to expect that the α/r 12
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approximateMethods - Physics 125c Course Notes Approximate...

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