1.3. Problems¶
1.1¶
An electron in a hydrogen atom finds itself in a state We neglect spin and the states are labeled as .

Normalise this state.

Calculate the probability of finding the electron with energy .

Calculate the probability of finding the electron with angular momentum component .

Calculate the probability of finding in a measurement.

The hydrogen atom is subject to a field which adds a term to the Hamiltonian. Give , if the state given above is the state at .
1.2.¶
Consider a spin1/2 particle in a spherically symmetric potential. The state of the particle is denoted . is the orbital angular momentum and the spin operator. The functions and are defined by where the second argument in the bravector on the righthand side denotes the spin and where with some given function of and the eigenfunctions of the angular momentum operators .

Which condition must be satisfied by in order for to be normalised?

A measurement of the spin component is performed. What are the possible results with respective probabilities? Same questions for and .

A value 0 is measured for the quantity . What is the state of the particle immediately after the measurement?
1.3.¶
An electron subject to a magnetic field in the direction evolves under the Hamiltonian At , the electron spin points along the positive direction.

Calculate the timedependent wavefunction.

Formulate the equations of motion of the expectation values of the spin components, , and .
From now, we consider the stationary behaviour rather than the dynamics.

Consider now a second electron. This second electron experiences a field in the direction, but with different magnitude. The two electrons interact via a weak transverse coupling. The total Hamiltonian is thus given by: where .
Use firstorder perturbation theory to estimate the energies and corresponding eigenstates for this Hamiltonian.

Use secondorder perturbation theory to estimate the energies and corresponding eigenstates for this Hamiltonian.

Find the exact solution for the energies and eigenstates.
1.4. Two spins¶
Consider two spin1/2 particles subject to a Heisenbergtype interaction where and are the spin operators for particle 1 and particle 2, respectively.

Find the energies and expand the corresponding eigenstates in the basis , where and denote the quantum numbers for the total momentum operators and , respectively.

Expand the eigenstates in the basis , where en are the quantum numbers of the operators en .

Consider now the modified Hamiltonian Give the exact energies and corresponding eigenstates.

Determine whether the ground state of the system is entangled. Note: an entangled state cannot be written as a product of states for particle 1 and particle 2.
1.5.¶
Consider two quantum dots in close proximity. A quantum dot is a small structure that can be occupied by an electron. We consider the case where only one state is available in each dot. In this problem, we assume that exactly one electron is present in the system. Denote the energies of the states as , where labels the dot. To these levels correspond the states , . As the dots are placed close together, they are coupled. The coupling constant is (the complex number) : where is the Hamiltonian.

Write the Hamiltonian of the system in the form of a matrix.

Find the spectrum of this Hamiltonian. Plot the spectrum as a function of for fixed . Also find the eigenfunctions.

Suppose we place an electron in dot 1 at . Give the time evolution of the wave function and show that the probability to find the electron in dot 2 as a function of time has the form Determine . Hint: write the quantum state as and use the spectrum found in (b) together with the timedependent solution of the Schrödinger equation.
1.6.¶
A particle is located at the origin of a line where the potential is zero. At , the particle is released. Find the wave function in the representation at time .
1.7.¶

Consider the three spin triplet states (in the standard notation ) and the singlet state , which are constructed out of two spin particles, A and B. The vector has the three Pauli spin matrices as its components, and is defined in a similar way. Show that and Obtain the eigenvalues of .

A system consisting of two spin particles is described by the Hamiltonian Assume , i.e. particle A has spin along the axis, and particle B has its spin oriented along the axis. Express in terms of the vector operator . Obtain for and determine the probability for finding the system in the states , and .
1.8.¶
We consider some properties of the Pauli matrices , and .

Show that , for .

Show that where whenever is an even permutation of and when is an odd permutation of .

For angular momentum quantum number , we have three possible states , where .\ Write down the matrix form of in the basis of these states.

Show that the matrices together with the matrix found in (c), satisfy the angular momentum commutation relations

Calculate the eigenvalues and eigenvectors of and .
1.9. Bloch
It may be useful to refer to the previous problem to refresh your knowledge concerning the Pauli matrices!
The state of a quantum twolevel system (also called quantum bit, qubit or Qbit) can be written as where the coefficients and are complex, and properly normalised: .

Explain why, without loss of generality, we can also write the state as with ranging from 0 to and ranging from 0 to .

There is a onetoone correspondence between the state and a unit vector in 3D, whose orientation is defined by polar and azimuthal angles and , respectively (see figure (1.1). Such a vector is called the Bloch vector associated with . Draw the Bloch vectors associated with the states , , , and .

Find , , and as a function of and . Here, , and are the Pauli matrices.

Show that .

How are , and related to the cartesian coordinates of the Bloch vector?

Unitary operations on one qubit correspond to rotations in the Bloch sphere. What kind of rotations correspond to the operators , and ? Specify the axis of rotation, and the rotation angle for each.

What does the operator do?

What does the operator do? Here, is a 3D unit vector.
(You only need to show that leaves the state with Bloch vector invariant.)
1.10. Gauge¶
The nonrelativistic Hamiltonian that describes the interaction of a charged particle with the electromagnetic field is

Assume that the wave function is changed by a constant phase Show that satisfies the original Schrödinger equation with the original vector potential .

Assume that the wave function is changed by a nonconstant phase Show that does not satisfy the original Schrödinger equation with the original vector potential .

Show that does satisfy the original Schrödinger equation but with a new vector potential where is a vector field. How is the gauge term related to the phase term ?\ How does the scalar potential change: