### Conceptual Questions

### 7.1 Wave Functions

What is the physical unit of a wave function, $\text{\Psi}(x,t)?$ What is the physical unit of the square of this wave function?

Can the magnitude of a wave function $({\text{\Psi}}^{*}\left(x,t\right)\phantom{\rule{0.2em}{0ex}}\text{\Psi}(x,t))$ be a negative number? Explain.

What is the physical meaning of a wave function of a particle?

### 7.2 The Heisenberg Uncertainty Principle

If the formalism of quantum mechanics is ‘more exact’ than that of classical mechanics, why don’t we use quantum mechanics to describe the motion of a leaping frog? Explain.

Can the de Broglie wavelength of a particle be known precisely? Can the position of a particle be known precisely?

Can we measure the energy of a free localized particle with complete precision?

### 7.3 The Schrӧdinger Equation

What is the difference between a wave function $\psi (x,y,z)$ and a wave function $\text{\Psi}(x,y,z,t)$ for the same particle?

Explain the difference between time-dependent and -independent Schrӧdinger’s equations.

Suppose a wave function is discontinuous at some point. Can this function represent a quantum state of some physical particle? Why? Why not?

### 7.4 The Quantum Particle in a Box

Using the quantum particle in a box model, describe how the possible energies of the particle are related to the size of the box.

Is it possible that when we measure the energy of a quantum particle in a box, the measurement may return a smaller value than the ground state energy? What is the highest value of the energy that we can measure for this particle?

For a quantum particle in a box, the first excited state $({\text{\Psi}}_{2})$ has zero value at the midpoint position in the box, so that the probability density of finding a particle at this point is exactly zero. Explain what is wrong with the following reasoning: “If the probability of finding a quantum particle at the midpoint is zero, the particle is never at this point, right? How does it come then that the particle can cross this point on its way from the left side to the right side of the box?

### 7.5 The Quantum Harmonic Oscillator

Is it possible to measure energy of $0.75\hslash \omega $ for a quantum harmonic oscillator? Why? Why not? Explain.

Explain the connection between Planck’s hypothesis of energy quanta and the energies of the quantum harmonic oscillator.

If a classical harmonic oscillator can be at rest, why can the quantum harmonic oscillator never be at rest? Does this violate Bohr’s correspondence principle?

Use an example of a quantum particle in a box or a quantum oscillator to explain the physical meaning of Bohr’s correspondence principle.

### 7.6 The Quantum Tunneling of Particles through Potential Barriers

When an electron and a proton of the same kinetic energy encounter a potential barrier of the same height and width, which one of them will tunnel through the barrier more easily? Why?

What decreases the tunneling probability most: doubling the barrier width or halving the kinetic energy of the incident particle?

Explain the difference between a box-potential and a potential of a quantum dot.

Can a quantum particle ‘escape’ from an infinite potential well like that in a box? Why? Why not?

A tunnel diode and a resonant-tunneling diode both utilize the same physics principle of quantum tunneling. In what important way are they different?