PrelExam

# PrelExam - PHY4211 Quantum Mechanics I Fall Term of 2005...

This preview shows pages 1–3. Sign up to view the full content.

PHY4211 Quantum Mechanics I Fall Term of 2005 Preliminary Examination (A Possible Set of Suggested Solution) Question 1: The time-independent Schrodinger equation is: () () 2 2 2 Vx E x m ψψ ⎡⎤ −∇ + = ⎢⎥ ⎣⎦ = , (1) and the potential is simply: () 0 2 else L x = . (2) Outside the well, the wave functions vanish; inside the well, they are given by: 2 2 2 0 mE xx ∇+ = = (3) Its solution is: 22 2 2 sin cos 0 sin cos 0s i n c o s 0 2 sin 0 2 cos 0 2 mE mE mE mE AL B L xA xB x Lm E m E L B L mE mE BL ψ ⎡⎤⎡⎤ += =+ ⎢⎥⎢⎥ ⎪⎣ ⎣⎦⎣⎦ ⎬⎨ ⎪⎪ ⎛⎞ = + = ⎜⎟ ⎝⎠ = = == = = , (4) Firstly it is possible that A=0 forever, then to obtain a nontrivial solution to the equation (1) we must require that B≠0 so that: () ( ) 2 2 cos 0 2 1 0,1, 2,3, 2 2 cos 2 1 mE L mE Ln n n x L π = ⇒= + =⋅ + = = , then the renormalized solution is easy to obtain: 2 cos 2 1 0,1, 2, xn x n LL = + = ⋅⋅⋅ . (5a) Similarly other solutions are: 2 sin 2 1,2,3, x n (5b) The corresponding eigen-states are respectively:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
() 2 22 2 12 0,1, 2, 2 En n mL π ⎡⎤ = + = ⋅⋅⋅ ⎢⎥ ⎣⎦ = ; (6a) 22 2 2 2 1, 2, n mL = = = . (6b) This problem could be also solved simply by shifting the range of the potential to, e.g. [0, 2L], and use the well known result from any introductory quantum mechanics textbook. This method will not be
This is the end of the preview. Sign up to access the rest of the document.

## PrelExam - PHY4211 Quantum Mechanics I Fall Term of 2005...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online