AP Physics 2 - Other Principles of Quantum Mechanics

The expectation value $\left< x \right>$ of a particle in a quantum system tells us what about the particle?

The most probable location of the particle

The exact location of the particle

The energy of the particle

The momentum of the particle

If the particle exists or not

Explanation

From a statistical standpoint, the expectation value of the position, $\left< x \right>$ , 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.

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 ).

Second excited state.

Ground state.

Fourth excited state.

Not enough information to solve the problem.

Explanation

The energy of the quantum system in the $n^{th}$ state is given by

$E_{n}=\frac{h^{2}}{8 m L^{2}}n^{2}$

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

$E_{photon}=h f=\frac{h c}{\lambda}$

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

$\frac{h^{2}}{8 m L^{2}}n^{2}=\frac{h c}{\lambda}$

$n^{2}=\frac{8 c m L^{2}}{h \lambda}$

$n=L \left ( \frac{8 c m }{h \lambda}\right )^{1/2}=3$

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

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

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 of the particle can be anything, positive or negative.

The energy must be negative and continuous.

The energy must be positive and continuous.

There is nothing constraining the energy of the particle.

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:

$E_n = -\frac{13.6eV}{n^2},n=1,2,3,...$

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

The square modulus of the wavefunction, given as $\left| \Psi \left(x,t \right ) \right|^2$ , contains what information about a particle in a quantum system?

$\left| \Psi \left(x,t \right ) \right|^2$ represents the probability distribution of the particle as a function of time

$\left| \Psi \left(x,t \right ) \right|^2$ represents the position of the particle as a function of time

$\left| \Psi \left(x,t \right ) \right|^2$ represents the square of the particle's position as a function of time

$\left| \Psi \left(x,t \right ) \right|^2$ represents the time-averaged position of the particle as a function of time

$\left| \Psi \left(x,t \right ) \right|^2$ represents the energy of the particle as a function of time

Explanation

By definition, $\left| \Psi \left(x,t \right ) \right|^2$ represents the probability distribution of a particle in a quantum system as a function of time. It is used to calculate the expectation value of other observables, such as position, momentum, current, angular momentum, just to name a few.

Other Principles of Quantum Mechanics

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