University College London Department Of Physics And Astronomy Phas3226 Quantum

UNIVERSITY COLLEGE LONDON DEPARTMENT OF PHYSICS AND ASTRONOMY PHAS3226 QUANTUM MECHANICS Problem Sheet 2 (2009) Solutions to be handed in by end of November 10 2009 The mark for each part is given in the right-hand margin

Question 1. A particle moving in one dimension is in a quantum state described by the wave-function: ψ(x) = A exp[ikx − γx2 ] (this kind of state is known as a ‘wave-packet’). Show that in order to ensure that this wave-function is normalised to unity, the constant A is given by the formula: |A| = (2γ/π)1/4 . (You may use the standard formula: Z

[3]

∞

2

e−αx dx = (π/α)1/2 . −∞

In the later parts of this question, you may wish to use the more general formulas: Z

∞

2n −αx2

x e −∞

(2n)! √ −(2n+1)/2 πα dx = n! 22n

and

Z

∞

2

x2n+1 e−αx dx = 0 . −∞

for any integer n ≥ 0 and any real constant α > 0.) What are the expectation values hxi and hpi of the position and momentum of the particle?

[4]

If many measurements of x are made, with the system always prepared in this same quantum
1/2 of the measured values? [5] state, what is the standard deviation ∆x ≡ hx2 i − hxi2 If many measurements of momentum p are made, what is the standard deviation ∆p ≡
1/2 hp2 i − hpi2 ? [5] Show that these values of ∆x and ∆p are related by: 1 ∆x∆p = h ¯. 2 [3]

PHAS3226/2009 Problem Sheet 2

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Question 2. In a simple harmonic oscillator, a particle of mass m in one dimension is acted on by a potential V (x) = 12 kx2 , where k is the spring constant. The frequency of the oscillator is ω = (k/m)1/2 . In the quantum treatment of the harmonic oscillator, it is helpful to work with step-up and step-down (creation and annihilation) operators a ˆ + and a ˆ− , defined as: a ˆ+ = a ˆ− =

i 1 h √ (mω/¯h)1/2 x ˆ − i(m¯hω)−1/2 pˆ 2 i 1 h √ (mω/¯h)1/2 x ˆ + i(m¯hω)−1/2 pˆ , 2

where x ˆ and pˆ are the position and momentum operators. The Hamiltonian of the system can then be expressed as: 1 ˆ = pˆ2 /2m + 1 kˆ x2 = h ¯ ω(ˆ a+ a ˆ− + ) . H 2 2 Show that in every energy eigenstate, the expectation values of the kinetic energy T and the potential energy V are equal: hT i = hV i. (Hint: express x ˆ and pˆ in terms of a ˆ + and a ˆ− by adding and subtracting the equations that define a ˆ + and a ˆ− . Then, use these expressions for x ˆ and pˆ to express Tˆ and Vˆ in terms of a ˆ+ and a ˆ− .) [8] Also, show that in the eigenstate n, whose energy is E n = (n + 21 )¯hω (n = 0, 1, . . .), we have: hx2 i1/2 hp2 i1/2 = cn h ¯, and obtain a formula for the constant c n .

[8]

From this, show that the ground state is a state of minimum uncertainty for x and p, with c0 = 1/2. [4]

PHAS3226/2009 Problem Sheet 2

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