HomeworkSet4

HomeworkSet4 - PHY4221 Quantum Mechanics (Fall Term Of...

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Unformatted text preview: PHY4221 Quantum Mechanics (Fall Term Of 2004) Problem Set 4 (Answers) Question 1: The calculation process is: [ ] [ ] 1 , 1 , d d xp xp dt dt xp xp xp t t t H H xp xp xp i t i xp H xp i t xp H i ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ = ∂ ∂ ∂ = + + ∂ ∂ ∂ ∂ = + + ∂ ∂ = + ∂ = h h h h (1) In above the last ‘=’ is due to the fact that neither x nor p depend on t explicitly. The Hamiltonian, of course, is given by: ( 29 2 2 p H V x m = + (2) This leads to: ( 29 2 2 2 2 2 2 2 3 2 3 2 2 2 2 1 1 , , 2 , , 2 2 2 d xp x x V dt i i x m x i i x x x V m x x x x x x V Vx m x x x x x x V m x x dV T x dx ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ∂ ∂ ∂ =- + ∂ ∂ ∂ ∂ ∂ ∂ =- ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =--- ÷ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =-- ∂ ∂ =- h h h h h h h h (3) In a stationary state, the LHS of equation (3) is zero, so we have: 2 dV T x dx = (4) Use this so-called Virial Theorem to the Harmonic Oscillator case we get: 2 2 2 2 1 2 2 2 d T x m x m x V dx ϖ ϖ = = = (5) This is what we are wanted to prove....
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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HomeworkSet4 - PHY4221 Quantum Mechanics (Fall Term Of...

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