Test on Quantum Mechanics: Motion and Spin

Question 1: Does the wavefunction for the first excited state of the harmonic oscillator have a node at x = 0?

A. Ano

B. Ne

Explanation: The wavefunction for the first excited state of the oscillator has a node at x = 0.

Question 2: According to the provided study materials, when a molecular bond vibrates from one energy level to the next due to the absorption of a photon, what is the relationship between the absorbed photon's frequency and the vibrational energy levels?

A. The photon's frequency is directly proportional to the total energy of the initial vibrational level.

B. The photon's frequency is equal to the separation between the initial and final vibrational energy levels, divided by Planck's constant (ħ).

C. The photon's frequency is inversely proportional to the square of the force constant (k) of the bond.

D. The photon's frequency is equal to the product of the mass of the vibrating atom and its displacement from equilibrium.

Explanation: The study materials state that if the vibration of a bond from one level to the next is caused by a photon, one would 'calculate the frequency... of the photon required.' It also states that 'Separation between two levels is: E +1 - E = ħ'. This implies that the energy difference E +1 - E is absorbed as a photon, and from the general relation E = h (where E is energy and is frequency), the frequency of the photon would be (E +1 - E)/ħ.

Question 3: For a particle on a ring, the angular momentum Jz is limited to specific values due to quantization.

A. Ano

B. Ne

Explanation: The study materials state that 'not all values of angular momentum (Jz) are permitted in quantum mechanics. Angular momentum and rotational energy are quantised.' It further explains that 'Only some wavefunctions have this property, thus, only some angular momenta are acceptable, thus only certain rotational energies exits – quantised' and later, 'the angular momentum is limited to the values: where ml has positive or negative values. Thus, Jz is quantised!'

Question 4: Is it possible to precisely know both the angular momentum (Jz) and the exact location (angle) of a particle on a ring at the same time?

A. Ano

B. Ne

Explanation: Angular momentum (Jz) and angle (θ) are complementary observables. Knowing angular momentum exactly eliminates the possibility of specifying a particle’s location, making its location completely indefinite.

Question 5: According to the study materials, what is the fundamental reason for the introduction of two quantum numbers to describe the motion of a particle on a sphere?

A. The particle's energy is quantized in two different directions.

B. The wavefunction for a particle on a sphere must satisfy two cyclic boundary conditions.

C. There are two different types of angular momentum for a particle on a sphere.

D. The Schrödinger equation in three dimensions requires two variables for its solution.

Explanation: The study materials explicitly state, 'The wavefunction of a particle on the surface of a sphere must satisfy two cyclic boundary conditions – resulting in two quantum numbers for its angular momentum.' This directly links the two quantum numbers to the two cyclic boundary conditions.