solk - Physics 741 Graduate Quantum Mechanics 1 Solution...

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

View Full Document Right Arrow Icon
Physics 741 – Graduate Quantum Mechanics 1 Solution Set K 1. [10] At t = 0, a single particle is placed in a harmonic oscillator 22 2 1 2 2 HP m mQ ω =+ in the superposition state ( ) 3 4 55 01 2 t Ψ= = + , That is, in a superposition of the first and second excited states. (a) [3] What is the wave function ( ) t Ψ at subsequent times? The wave function has been written in terms of eigenstates of the Hamiltonian, so this makes it relatively easy. The energy of the state n is ( ) 1 2 n + = , and therefore the state will evolve as () 32 52 3 2 5 2 33 44 5 5 121 2 it i t i t te e e e ωω ψ −−− = + == . (b) [7] What are the expectation values Q and P at all times? These are most easily calculated using the raising and lowering operators ( ) ( ) 5 5 5 5 3 42 4 2 5 2 21 2 1 2 2 0 2 2 2 1 3 3 c o s , Qm t a a t me e a ae e e e e e m t ωψ −− + + + + + =⋅ + = = = = and ( ) ( ) 5 5 5 5 3 4 2 5 2 2 1 2 2 2 20 3 3 2 1 s i n . Pim t aa t im e e e e e e e e e e m t =− + + = = = =
Background image of page 1

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

View Full DocumentRight Arrow Icon
2. [15] In class we assumed that the coupled harmonic oscillators all had the same mass. Consider now the case where the oscillators have different masses, so that 2 1 2 2 ii i j i j j HP m k Q Q =+ ∑∑ (a) [5] Rescale the variables P i and X i to new variables ˆ i P and ˆ i Q with the usual commutation relations: ˆ ˆ , ij i j QP i δ ⎡⎤ = ⎣⎦ = such that in terms of the new variables, the Hamiltonian is given by 2' 1 2 ˆˆ ˆ 2 j i j j m k Q Q where m is an arbitrary mass that you can choose.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/15/2011 for the course PHYS 742 taught by Professor Unknow during the Spring '04 term at Harvard.

Page1 / 4

solk - Physics 741 Graduate Quantum Mechanics 1 Solution...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online