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# 8 Quiz

Answer the following questions in order to test your understanding of the key ideas that you have been learning about.

## Question 1

Which of the following statements about eigenvalues, eigenstates, eigenvectors and eigenfunctions are true?

An eigenfunction is special type of function that remains essentially unchanged (except for a scaling factor) when acted upon by a given linear operator.True
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.True
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.True
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.True
An eigenvector of a square matrix is a nonzero vector that gets scaled by a certain value when the matrix is applied to it. True
There is always only one eigenvalue and corresponding eigenfunction associated with each eigenvalue equation False

### 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

What are the eigenvalues and eigenvectors of the following matrix?

For eigenvalue the eigenvector is and for eigenvalue the eigenvector is
For eigenvalue the eigenvector is and for eigenvalue the eigenvector is
For eigenvalue the eigenvector is and for eigenvalue the eigenvector is
For eigenvalue the eigenvector is and for eigenvalue the eigenvector is

### 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

If a general spin state is written as , which of the following statements is true?

The probability of a measurement indicating spin-up is
The probability of a measurement indicating spin-down is
The probability of a measurement indicating spin-down is
The probability of a measurement indicating spin-down is
The probability of a measurement indicating spin-up is

### 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.

symmetric not entangled state

symmetric entangled state

antisymmetric entangled state

Using the following two lists, match each numbered item with the correct letter.

1. 1

   symmetric not entangled state

2. 2

   symmetric entangled state

3. 3

   antisymmetric entangled state

The correct answers are:

- 1 =
- 2 =
- 3 =

### 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

The quantum NOT gate is represented by which of the following matrices?

### Answer

The quantum NOT gate is represented by the Pauli-X operator,

## Question 6

The Hadamard gate is represented by which of the following matrices?

### Answer

The Hadamard gate is represented by

## Question 7

Match the following quantum gates with the correct result.

NOT gate

Flips the state of a qubit

CNOT gate

Entangles a pair of disentangled qubits

Hadamard gate

Transforms a qubit into a superposition state

Using the following two lists, match each numbered item with the correct letter.

1. NOT gate

2. CNOT gate

3. Hadamard gate

1. 1

   Flips the state of a qubit

2. 2

   Entangles a pair of disentangled qubits

3. 3

   Transforms a qubit into a superposition state

The correct answers are:

- 1 =
- 2 =
- 3 =

### 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

If an input qubit is passed to a Hadamard gate, and the output from the Hadamard gate is then passed as input to another Hadamard gate, what will be the output from the second Hadamard gate?

### 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.

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