Elementary Examples of Schrödinger Picture

In this section, we will explore a few elementary examples of the Schrödinger picture that we introduced in the previous section. We will start with some physical examples like spin 1/2 particles and neutrinos, and then we will conclude by deriving the energy-time uncertainty relation .

Table of Contents

• Table of Contents
• Spin 1/2 Particle in a Magnetic Field
  • Example 1: +z State
  • Example 2: +x State
• Neutrino Oscillation

Spin 1/2 Particle in a Magnetic Field

In this example, we will consider a spin 1/2 particle subjected to a magnetic field. It has a magnetic moment and the magnetic field is assumed to be uniform and pointing in the direction. The Hamiltonian for this system is given by:

The Hamiltonian is a scalar product of the magnetic field and the spin operator . Since is a multiple of , naturally commutes with . As such, we can use the eigenstates of as our basis states. The eigenstates of are given by:

If we introduce the variable , then the energy difference between the eigenstates is , and the Hamiltonian can be written as:

Then, the time-evolution operator becomes:

To see how this operator acts on the state vector, we can write it in terms of the eigenstates of ; . Since and , we have:

This means that the state vector is a superposition of the two eigenstates and , with each eigenstate picking up a phase factor that oscillates in time. We will now apply the time-evolution operator to a few different states.

Example 1: +z State

Suppose we have a spin 1/2 particle in the state at time . The ket has coefficients and . Because the state is an eigenstate of , it is an energy eigenket and is hence stationary (time-independent). We can see this more clearly by applying the time-evolution operator to the state:

This means that the state vector is simply a phase factor times the original state vector . The expectation value of any observable that is compatible with will be constant in time. (Recall that the magnitude of a state vector does not matter in the context of quantum mechanics; only the direction matters.)

Example 2: +x State

Suppose we have a spin 1/2 particle in the state at time . Recall from previous discussions that the state can be written in terms of the basis states as:

Hence has coefficients and . We can apply the time-evolution operator to the state:

Applying the Born rule, we can find the probability of measuring the state at time :

Recall and . Using this, we can rewrite the above equation depending on the sign of the second term:

And the expectation value of the observable is given by:

where we have first used the definition of the expectation value from probability; , and then used the trigonometric identity . This means that the expectation value of oscillates in time with a frequency . Physically, this is because when a spin 1/2 particle in the state is subject to a magnetic field in the direction, the field causes the spin to precess about the axis. This is similar to how a top precesses about the vertical axis when it is spinning.

To be sure, we can see that this matches Born's frequency condition, . The energy difference between the two eigenstates is , and so the frequency of oscillation is , which matches our result.

Neutrino Oscillation

Before the 1930s, when beta decay was being studied, it was thought that it only involved the decay of a neutron into a proton and an electron. However, it was found that the decay of a neutron into a proton and an electron was not possible without violating conservation of energy. This is because the neutron is heavier than the proton and electron combined. Wolfgang Pauli proposed the existence of a new particle, which he called the "neutrino", to explain this discrepancy. In 1956, Clyde Cowan and Frederick Reines experimentally confirmed the existence of the neutrino.

The neutrino is a neutral particle with a very small mass, and it interacts very weakly with matter. Each lepton (electron, muon, and tau) has a corresponding neutrino (electron neutrino, muon neutrino, and tau neutrino). These are called the different flavors of neutrinos. The neutrino is a fermion, and it has a spin of 1/2.

For simplicity we will consider two neutrinos, the electron neutrino and the muon neutrino . These are the eigenstates of the Hamiltonian for the interaction of the neutrinos with the weak force. Neutrinos also have other interactions in which their eigenstates are the mass eigenstates and . To convert between flavor and mass eigenstates, we can use the following transformation:

In other words, we use a simple rotation with a mixing angle to convert between the two bases.