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Physics 451 Quantum mechanics Fall 2012 Karine Chesnel Phys 451 Announcements Test 1 next week Mo Sep 24 – Th Sep 27 • Today: Review - Monday: Practice test Be prepared to present the solution of your chosen problem during class (~ 5 to 10 min) Phys 451 EXAM I • Time limited: 3 hours • Closed book • Closed notes • Useful formulae provided Review lectures, Homework and sample test Phys 451 EXAM I 1. Wave function, probabilities and expectation values 2. Time-independent Schrödinger equation 3. Infinite square well 4. Harmonic oscillator 5. Free particle Phys 451 Review I What to remember? Review I Quantum mechanics 1. Wave function and expectation values Density of probability ( x, t ) ( x, t ) Normalization: ( x, t ) 1 2 2 Review I Quantum mechanics 1. Wave function and expectation values p i dx x x xdx * * “Operator” x “Operator” p Q Q x, i dx x * Quiz 9a What is the correct expression for the operator T= Kinetic energy? A. 2m x B. x 2 / 2m C. 1 2m x 2 2 2 2 1 D. m 2 x 2 2m x E. 2 2m x 2 2 Review I Quantum mechanics 1. Wave function and expectation values Variance: x x x 2 2 p p p 2 2 x p 2 2 2 Heisenberg’s Uncertainty principle Review I Quantum mechanics 2. Time-independent Schrödinger equation 2 2 i V 2 t 2m x Here V ( x) The potential is independent of time General solution: ( x, t ) ( x) (t ) “Stationary state” Review I Quantum mechanics 2. Time-independent Schrödinger equation 2 1 d 1 d 2 i V 2 dt 2m dx Function of time only (t ) e Solution: iE E Function of space only t ( x, t ) ( x)eiEt / Stationary state Quantum mechanics Review I 2. Time-independent Schrödinger equation Q for each Stationary state is independent of time p m v m d x 0 dt ^ H E A general solution is ( x, t ) cn n ( x, t ) n 1 where n ( x, t ) n ( x)eiEnt / ^ H cn En 2 n Review I Quantum mechanics 3. Infinite square well d 2 2 k 2 dx with 0 a The particle can only exist in this region x k 2mE Review I Quantum mechanics 3. Infinite square well 3 , E3 n 2 n sin a a x Excited states 2 , E2 Quantization of the energy Ground state 0 1 , E1 a x n 2 2 2 En 2 2ma Review I Quantum mechanics 3. Infinite square well ( x, t ) cn n ( x)e n 1 iEn t / x n 1 a cn sin(n )e iEnt / 2 n x cn n* ( x) ( x, 0)dx sin( ) ( x, 0)dx a a ^ H n En n * n nm m ^ H cn En n 1 2 Quiz 9b The particle is in this sinusoidal state. What is the probability of measuring the energy E0 in this state? A. 0 B. 1 C. 0.5 D. 0.3 0 a x E. 1 9 Review I Quantum mechanics 4. Harmonic oscillator V(x) V ( x) 1 2 1 kx m 2 x 2 2 2 a 1 2m ip m x • Operator position • Operator momentum 1 H a a 2 x x 2m pi or m 2 a a a a 1 H a a 2 Review I Quantum mechanics 4. Harmonic oscillator Ladder operators: a n 1 Raising operator: a n n 1 n1 Lowering operator: a n n n1 n 1 n a 0 n! 1 En n 2 a n n1 En En En Quantum mechanics Review I 5. Free particle p2 E 2m d 2 2 k 2 dx ( x, t ) ( x ) e 1 ( x, t ) 2 k with iEt / Ae ( k )e i kx t i ( kx t ) dk 2mE Be i kx t Wave packet Quantum mechanics Review I Free particle Method: 1. Identify the initial wave function ( x, 0) 2. Calculate the Fourier transform 1 (k ) 2 ( x, 0)e ikx dx 3. Estimate the wave function at later times 1 ( x, t ) 2 i ( kx t ) ( k ) e dk Quantum mechanics Review I 5. Free particle 1 ( x, t ) 2 Dispersion relation (k ) i ( kx t ) ( k ) e dk here k2 2m here Physical interpretation: • velocity of the each wave at given k: • velocity of the wave packet: v phase vgroup k d dk v phase k 2m vgroup k m