# AP Physics 2 : Other Principles of Quantum Mechanics

## Example Questions

### Example Question #1 : Other Principles Of Quantum Mechanics

A proton is confined to a one-dimensional box of length . It has an energy equal to that of a photon with a wavelength of . What excited state is the proton in? (Remember, the first excited state is where  since the ground state is ).

Fourth excited state.

Not enough information to solve the problem.

Second excited state.

Ground state.

Second excited state.

Explanation:

The energy of the quantum system in the  state is given by

where  is Planck's constant,  is the mass of the proton and  is the length of the box. The energy of a photon is given by

where  is the frequency,  is the speed of light and  is the wavelength. Setting these equal we can solve for ,

Since the ground state is , the proton must be in the second excited state.

### Example Question #1 : Other Principles Of Quantum Mechanics

The expectation value  of a particle in a quantum system tells us what about the particle?

The energy of the particle

The exact location of the particle

If the particle exists or not

The most probable location of the particle

The momentum of the particle

The most probable location of the particle

Explanation:

From a statistical standpoint, the expectation value of the position, , can only tell us the most probable location of the particle. A central idea in quantum mechanics is that we can never really know exactly where a particle is as a function of time, but rather where we are most likely to find the particle if we choose to observe it.

### Example Question #3 : Other Principles Of Quantum Mechanics

If a particle in a quantum system is bounded, this means what about the calculated particle energy?

The energy of the particle can be anything, positive or negative.

The energy must be positive and continuous.

The energy is described by discrete energy levels, and the particle can only have energy that corresponds to these levels, positive or negative.

The energy must be negative and continuous.

There is nothing constraining the energy of the particle.

The energy is described by discrete energy levels, and the particle can only have energy that corresponds to these levels, positive or negative.

Explanation:

A key characteristic of bound-state systems is the quantization of the energy into discrete energy levels. The energy can be negative or positive, but the particle can only have an energy that corresponds to one of these energy levels, nothing other. An example is the energy of a hydrogen atom, where the energy levels are given by:

These discrete numbers come from the energy of an electron, which is the fundamental charge.

### Example Question #4 : Other Principles Of Quantum Mechanics

The square modulus of the wavefunction, given as  , contains what information about a particle in a quantum system?

represents the square of the particle's position as a function of time

represents the probability distribution of the particle as a function of time

represents the energy of the particle as a function of time

represents the time-averaged position of the particle as a function of time

represents the position of the particle as a function of time