Question 1

Why do all bound particles in quantum mechanics have only a discrete set of allowed energies?

Accepted Answer

Because the wavefunction must satisfy boundary conditions, leading to quantization.

Question 2

For a mass $$m on a $$spring with spring constant $$k$$, the natural frequency of oscillation is given by $$\omega = \sqrt{\frac{k}{m}}. $$What happens to $$\omega if$$ $$k is $$increased while $$m is $$kept constant?

Accepted Answer

$$\omega$$ increases.

Question 3

In the quantum harmonic oscillator, why are the energy levels evenly spaced with spacing $$\hbar \omega$$?

Accepted Answer

Because the potential is quadratic, leading to a linear operator algebra.

Question 4

What is the minimum energy of a quantum harmonic oscillator, and what principle explains its existence?

Accepted Answer

$$\frac{1}{2}\hbar \omega$$, explained by the Heisenberg Uncertainty Principle.

Question 5

For a particle in a box of length $$L$$, what is the minimum energy and how is it related to the size of the box?

Accepted Answer

$$E_{min} \propto \frac{h^2$}{mL^2$}$$, due to the maximum allowed wavelength and the uncertainty principle.

Question 6

In the context of the hydrogen atom, using the uncertainty principle, what is the estimated ground state energy proportional to?

Accepted Answer

$$-\frac{me^4$}{2\$$hbar^2$$}$$

Question 7

Why are the harmonic oscillator wavefunctions not zero at the classical turning points?

Accepted Answer

Because quantum particles can tunnel into classically forbidden regions.

Question 8

What is the basic nature of a wavefunction in a classically forbidden region?

Accepted Answer

It decays exponentially.

Question 9

Why do harmonic oscillator wavefunctions spread out as the energy level increases?

Accepted Answer

Because higher energy states have larger amplitudes and the particle explores a wider region.

Question 10

Why do harmonic oscillator wavefunctions have shorter wavelength near $$x=0$$ and longer wavelength near the classical turning points?

Accepted Answer

Because the kinetic energy is higher near $$x=0$$, leading to higher momentum and shorter wavelength.

Question 11

For large energy levels in the harmonic oscillator, why is the amplitude of the wavefunction smaller near $$x=0$$ and larger near the classical turning points?

Accepted Answer

Because the classical speed is higher near $$x=0$$, so the probability density is lower.

Question 12

Why is the classical probability function for a harmonic oscillator proportional to $$1/v$$, where $$v is $$the classical speed?

Accepted Answer

Because the particle spends more time where its speed is lower, increasing probability density.

Question 13

In what sense is the ground state of the quantum harmonic oscillator the 'least classical' state?

Accepted Answer

Because its probability distribution is concentrated near $$x=0$$, unlike the classical distribution.

Question 14

The $$x-$$dependent part of the harmonic oscillator wavefunction is $$\psi_n(x) = [n	ext{th order polynomial in } x] 	imes \exp\left(-\frac{1}{4}\left[\frac{x}{A_0$}ight]^2ight). $$What does the exponential term represent?

Accepted Answer

The tunneling aspect of the wavefunction, decaying in classically forbidden regions.

Question 15

Using dimensional analysis, why is there no combination of $$k$$ and $$m in $$classical physics that has the dimensions of length, but in quantum physics, a unique length scale $$A_0$$ exists?

Accepted Answer

Because quantum physics introduces $$\hbar$$, allowing a combination $$A_0^2$$ = \hbar/(2\sqrt{km})$$ with dimensions of length.

Question 16

What determines the 'scale of size' of the quantum oscillator, analogous to the Bohr radius for the atom?

Accepted Answer

The length scale $$A_0$$ which depends on $$\hbar$$, $$k$$, and $$m.$$

Question 17

Which property of the harmonic oscillator potential is responsible for the evenly spaced energy levels?

Accepted Answer

Its quadratic form.

Question 18

If the energy of a harmonic oscillator is $$E$$, what is the amplitude $$A of $$oscillation in terms of $$E$$, $$k$$, and $$m$$?

Accepted Answer

$$A = \sqrt{\frac{2E}{k}}$$

Question 19

For the quantum harmonic oscillator, what is the spacing between adjacent energy levels?

Accepted Answer

$$\hbar \omega$$

Quantum Harmonic Oscillator: Energy Levels, Wavefunctions, and Quantum Intuition

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