Harvard University
Physics 143b: Quantum Mechanics II
Practice Final exam
1. We refer to the states of a qubit using the eigenstates |↑i,|↓i of σz, which we write as
|0i,|1irespectively.
(a) Two qubits are initialized in the state |0,0i. A Hadamard operation is applied on
the first qubit, followed by a cNOT operation (with the first qubit as the control
qubit); see definitions below. What is the final quantum state of the qubits ?
(b) After the operations in (a), Alice takes the first qubit, and Bob takes the second
one. Alice then applies a magnetic field Bon her qubit in the zdirection for
a time T(this is described by the Hamiltonian HAlice =−Bσz). Finally Alice
measures the value of σxon her qubit and finds the value +1. What is the state
of Bob’s qubit after this measurement ?
(c) Suppose Alice had performed the operations in (b) in the opposite order i.e. first
measured σxeigenvalue +1, and then applied the magnetic field. What would
then be the state of Bob’s qubit at the end ?
Useful info:
•The Hadamard operation His defined by H|0i= (|0i+|1i)/√2
and H|1i= (|0i−|1i)/√2.
•The cNOT operation, with first qubit the control qubit, is defined by C|00i=
|00i,C|01i=|01i,C|10i=|11i, and C|11i=|10i.
2. A non-relativistic, spinless particle of mass mis placed in a box with potential V(x)=0
for 0 < x < L. We will have V(x) = ∞for x < 0 and x>Lfor all times t.
(a) What are the energies, En, and eigenfunctions ψn(x) ? Here n= 1,2,3,4. . ..
(b) At time t= 0, the particle is in the ground state ψ1(x). For times 0 < t < T, the
potential is changed to
V(x) = (0,0< x < L/2
V0, L/2< x < L ,0< t < T, (1)
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