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Transcript
Quantum Mechanics I: Quiz Total: 15 marks NOTE: The harmonic oscillator creation-annihilation operators are q ip a† = mω (x − mω ) and its Hermitian conjugate: they satisfy [a, a† ] = 1. 2h̄ √ √ The action on the states is a|ni = n|n − 1i, a† |ni = n + 1|n + 1i. (1) From the Heisenberg equation of motion, find the time evolution equation for the operator V (x) = 12 mω 2 x2 corresponding to the potential energy (using the expression for product operator commutators). [4 mks] (2) Say the harmonic oscillator is in the ground state |0i initially. It is subjected to a sharp impulse-like interaction that switches on for a very short time ∆t and then switches off. The Hamiltonian for this interaction is Hint = λx4 , where x is the position coordinate. (a) Expand out the time evolution operator Uint (t) = e−iHint t to linear order for this impulse interaction. This is a good approximation since ∆t is very small (and the interaction strength λ is also assumed to be small). [3 mks.] (b) Evaluate carefully the time evolved state after the interaction has switched off using the evolution operator in (a), using the expression for the position operator in terms of the creation-annihilation operators. [5 mks] (c) Calculate the probability that the system, after the impulse interaction has switched off, is in the excited state |ni. For nonzero probability, how large can n be under the assumptions and approximations of (a), (b) ? What happens when the time ∆t of interaction is increased ? [3 mks] 1
