Final

171.303 Fall 2010(open book and notes)

1. (20 points) Three spin-1/2 particles are released in a pure quantum state of totalspin quantum number of s = 3/2 and total z-component of angular momentumm = +1/2, labeled |3/2,+1/2〉.(a) Write this state as a linear combination of states labeled by the z-components

of the individual particle spins, of the form | ±z〉1| ±z〉2| ±z〉3. The subscriptslabel particles 1, 2, and 3.

(b) If particle 1 is measured to be in the |+x〉 state, what is the probability thatparticles 2 and 3 are together in a zero total angular momentum state?

(c) If particle 1 is measured to be in the |+x〉 state, and particle 2 is also measuredto be in the | +x〉 state, what is the probability that particle 3 will be foundin the |+z〉 state?

(d) If particle 1 is measured to be in the |+x〉 state, and particle 2 is measured tobe in the | −x〉 state, what is the probability that particle 3 will be found inthe |+z〉 state?

2. (35 points) A particle of mass m is in the ground state |ψ〉 of a harmonic oscillatorpotential. The potential suddenly increases its natural frequency by a factor 4.Thus, while the new potential is V (x) = 1

2mω2x2, the initial wave function of the

particle is that of the ground state of a harmonic oscillator with frequency ω/4. Interms of raising and lowering operators, the new Hamiltonian is H = hω(a†a+1/2).

(a) What is the old ground state wave function 〈x|ψ〉 in terms of ω and m.

(b) Find the probability that a measurement will find the particle in the groundstate of the new harmonic oscillator.

(c) What linear combination of the operators x and p annihilates the state |ψ〉?Call that linear combination b. Thus, b|ψ〉 = 0. Hint, what is the loweringoperator of the old oscillator?

(d) Write b in terms of raising and lowering operators a† and a.

(e) Label the states of the new oscillator as |n〉, and its energies as En. Whatis the probability that the particle will be measured to have energy E1? E2?To compute these, use 〈0|b|ψ〉 = 0 and 〈1|b|ψ〉 = 0 to relate them to theprobability of being in the ground state (n = 0) .

(f) Bonus: write a recursion relation to calculate all probability amplitudes 〈n|ψ〉.

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3. (15 points) A particle with mass m moves in the following potential:

V (x) =

∞ x < 00 0 < x < a

V0(x/a− 1) x > a

(a) Sketch an estimate of the solution to the Schrodinger equation based solely onthe qualitative features of the potential.

(b) Use the WKB approximation to obtain an approximate expression for boundstate energies.

(c) Check your result by comparing the appropriate limit of it to the bound stateenergies for the infinite square well potential.

4. (30 points) A particle of mass m is in the ground state |ψ〉 of a one-dimensionalharmonic oscillator potential when the potential suddenly shifts its position by anamount ∆. The result is a harmonic oscillator, H = hω(a†a + 1/2), whose particleis in an initial state 〈x|ψ〉 = ψ(x) = ψ0(x−∆), shifted from the ground state wavefunction ψ0(x).

(a) Compute the probability of finding the particle in the ground state, |〈0|ψ〉|2.

(b) |ψ〉 can be written as the ground state shifted by the translation operator,T (∆)|0〉, with T (∆) = e−ip∆/h. Write the translation operator in terms ofraising and lowering operators.

(c) Find[a, T (∆)

]and

[a†, T (∆)

](remember the tricks we used for coherent states

in class).

(d) Find the expectation value of the energy in this state, 〈ψ|H|ψ〉 and see that it isequivalent to the classical energy of a particle in a harmonic oscillator shifted adistance ∆ from the center plus the zero point energy of the harmonic oscillator.

(e) Compute the probability of finding the particle in any excited state, |〈n|ψ〉|2,and then write |ψ〉 as a linear combination of energy eigenstates.

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