8 Quiz
Answer the following questions in order to test your understanding of the key ideas that you have been learning about.
Question 1
a.
An eigenfunction is special type of function that remains essentially unchanged (except for a scaling factor) when acted upon by a given linear operator.
b.
An eigenstate in quantum mechanics is a special quantum state that remains unchanged, except for a multiplicative factor, when a specific quantum operator acts on it.
c.
An eigenvalue is a special scalar associated with a linear transformation of a square matrix. It represents how much a given vector is scaled when that matrix is applied to it.
d.
An eigenstate is a state for which the outcome of a measurement of a certain observable (like energy, position, or momentum) will always yield a specific, definite value.
e.
An eigenvector of a square matrix is a nonzero vector that gets scaled by a certain value when the matrix is applied to it.
f.
There is always only one eigenvalue and corresponding eigenfunction associated with each eigenvalue equation
The correct answers are a, b, c, d and e.
a.
True
b.
True
c.
True
d.
True
e.
True
Answer
The first five statements are all true. The last one is false: there may be more than one eigenvalue and corresponding eigenfunction associated with each eigenvalue equation.
Question 2
a.
For eigenvalue the eigenvector is and for eigenvalue the eigenvector is
b.
For eigenvalue the eigenvector is and for eigenvalue the eigenvector is
c.
For eigenvalue the eigenvector is and for eigenvalue the eigenvector is
d.
For eigenvalue the eigenvector is and for eigenvalue the eigenvector is
The correct answer is b.
Answer
Following the prescription described in the course: , , and . So we first need to solve the quadratic equation
which is simply
This can be written as
So it has solutions and . These are the two eigenvalues.
We now write the two eigenvector equations:
For eigenvalue these reduce to
Both equations imply that , so and and the first eigenvector is
For eigenvalue these reduce to
Both equations imply that , so and and the second eigenvector is
Question 3
a.
The probability of a measurement indicating spin-up is
b.
The probability of a measurement indicating spin-down is
c.
The probability of a measurement indicating spin-down is
d.
The probability of a measurement indicating spin-down is
e.
The probability of a measurement indicating spin-up is
The correct answer is b.
Answer
The probability of the outcome of a measurement indicating spin-up is and for spin-down is .
Question 4
Match the following two-particle spin states with the correct descriptions.
Using the following two lists, match each numbered item with the correct letter.
Match each of the previous list items with an item from the following list:
a.symmetric not entangled state
b.symmetric entangled state
c.antisymmetric entangled state
- 1 = a,
- 2 = b,
- 3 = c
Answer
The triplet states (i.e. and and ) are symmetric and the singlet state (i.e. ) is antisymmetric under particle exchange. Two-particle states which cannot be factorised (i.e. and ) are known as entangled states. The other states (i.e. and ) are not entangled.
Question 5
a.
b.
c.
d.
e.
The correct answer is c.
Answer
The quantum NOT gate is represented by the Pauli-X operator,
Question 6
a.
b.
c.
d.
e.
The correct answer is d.
Answer
The Hadamard gate is represented by
Question 7
Match the following quantum gates with the correct result.
Using the following two lists, match each numbered item with the correct letter.
NOT gate
CNOT gate
Hadamard gate
Match each of the previous list items with an item from the following list:
a.Transforms a qubit into a superposition state
b.Flips the state of a qubit
c.Entangles a pair of disentangled qubits
- 1 = b,
- 2 = c,
- 3 = a
Answer
A NOT gate flips the state of a qubit. A CNOT gate can entangle a pair of disentangled qubitts. A Hadamard gate can transform a qubit into a superposition state.
Question 8
a.
b.
c.
d.
e.
f.
The correct answer is b.
Answer
The circuit can be written as . The action of the first Hadamard gate produces a superposition state:
Then passing this through the second Hadamard gate we have
The action of the second Hadamard gate is therefore to restore the original input qubit.