Chapter 8 The Variation Method 1 8 1

Chapter 8 The Variation Method 1

8. 1 The Variation Theorem • The variation method is one of the approximation methods needed to deal with time-independent Schrodinger equation for systems that contain interacting particles. • It allows us to approximate the ground-state energy of a system without solving the Schrodinger equation. 2

The variation theorem Given a system: 1. The Hamiltonian operator is time independent. 2. The lowest-energy eigenvalue is E 0. 3. If is any normalized, well-behaved function, and it satisfies the boundary conditions of the system, then We can calculate an upper bound for the system’s ground-state energy. 3

prove (8. 1) 4

prove (8. 1) 5

prove (8. 1) 6

prove (8. 1) 7

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Trial variation function • is called a trial variation function, called the variation integral. is We try many trial functions to arrive at a good approximation to E 0. ◎ The lower the value of the integral, the better the approximation we have to E 0. ◎ 9

Trial variation function 10

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8. 2 Extension of the Variation Method 13

Extend the variation method to the first excited state: 14

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8. 3 Determinants 16

The value of the first, second and third order determinants 17

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8. 4 Simultaneous linear equations 28

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the difficulty • If the determinant of the inhomogeneous system of n 1 equations in n-1 unknowns is zero, then Cramer’ rule has a zero in the denominator and is of no use. • Solutions: 1. by assigning the arbitrary value to another of the unknowns rather than to xn. 2. by discarding some other equation, rather than the last one. 33