TRB PG Assistant - Schrödinger wave equation and model systems: particle in a box, rigid rotator and harmonic oscillator Question Bank with Options

1. The energy of the first excited state (n=2) of a particle in a 1D box of length L is:

A) h²/8mL²

B) 4h²/8mL²

C) 9h²/8mL²

D) π²h²/2mL²

2. Which model system best approximates the vibrational motion of a chemical bond in a molecule?

A) Particle in a box

B) Rigid rotator

C) Harmonic oscillator

D) Free particle

3. The Schrödinger equation is a postulate of quantum mechanics. Its solutions (wave functions) must be:

A) Discontinuous and non-unique

B) Continuous, single-valued, and normalizable

C) Arbitrary functions

D) Complex and non-integrable

4. If the force constant 'k' of a harmonic oscillator increases, the vibrational frequency (ω = √(k/m)) will:

A) Increase

B) Decrease

C) Remain unchanged

D) Become zero

5. The radial part of the Schrödinger equation for the hydrogen atom involves a potential term. For a rigid rotator, the potential energy is:

A) Zero

B) Constant

C) Dependent on angle

D) Dependent on distance

6. For a particle confined to a 1D box, the energy is quantized. This quantization arises from:

A) The mass of the particle

B) The length of the box and the wave nature of the particle

C) The charge of the particle

D) The temperature of the system

7. The energy levels of a quantum harmonic oscillator are degenerate only if:

A) The force constant is zero

B) The mass is zero

C) The system is in more than one spatial dimension

D) The quantum number v is zero

8. In the harmonic oscillator model, the potential energy function is:

A) V(x) = constant

B) V(x) = ½kx²

C) V(x) = ∞ for x<0 and x>L

D) V(x) = 0

9. What is the physical meaning of the quantum number 'm_J' for a rigid rotator?

A) The magnitude of the total angular momentum

B) The projection of the angular momentum onto the z-axis

C) The rotational energy level

D) The moment of inertia

10. The Schrödinger equation for a free particle (V=0) simplifies to:

A) Ĥψ = 0

B) Kinetic Energy Operator ψ = Eψ

C) Ĥψ = -ħ²/2m ∇²ψ

D) Ĥψ = ħ²/2m ∇²ψ

11. The shape of the probability distribution for a harmonic oscillator in a high energy state (large 'v') approaches that of a:

A) Particle in a box

B) Classical harmonic oscillator

C) Rigid rotator

D) Free particle

12. A particle in a 1D box of length L has a wave function ψ(x) = √(2/L)sin(3πx/L). What is the quantum number 'n' for this state?

A) 1

B) 2

C) 3

D) 6

13. If the reduced mass of a diatomic molecule increases, while the bond length remains the same, how does its rotational constant (B = ħ²/2I) change?

A) Increases

B) Decreases

C) Remains the same

D) Becomes zero

14. The selection rule for transitions between rotational energy levels in a rigid rotator, observable via microwave spectroscopy, is:

A) ΔJ = ±1

B) ΔJ = ±2

C) ΔJ = 0

D) ΔJ = ±1, ±2

15. The selection rule for transitions between vibrational energy levels in a harmonic oscillator, observable via infrared spectroscopy, is:

A) Δv = ±1

B) Δv = ±2

C) Δv = ±3

D) Δv = 0

16. For a rigid rotator, the angular momentum quantum number 'J' can take values:

A) 0, 1, 2, ...

B) 1, 2, 3, ...

C) 0, ±1, ±2, ...

D) ±1, ±2, ±3, ...

17. What is the role of the commutation relation [Ĥ, ψ] = 0 in quantum mechanics?

A) It implies that energy and the wave function are not measurable quantities

B) It implies that energy and the wave function can be simultaneously known with arbitrary precision

C) It implies that the wave function is an eigenfunction of the Hamiltonian

D) It leads to the uncertainty principle

18. The Schrödinger equation is a differential equation that describes how the quantum state of a physical system changes over time. The time-independent form is used when:

A) The potential energy is time-dependent

B) The total energy of the system is conserved and the potential energy is time-independent

C) The system is undergoing random fluctuations

D) We are interested in the rate of change of probability

19. What happens to the probability density of a particle in a 1D box as 'n' increases?

A) It becomes more localized

B) It becomes more spread out and complex with nodes

C) It remains uniform across the box

D) It becomes zero

20. Consider a particle in a 1D box. As 'n' increases, the energy levels:

A) Get closer together

B) Get farther apart

C) Remain equally spaced

D) Become degenerate

21. If a harmonic oscillator has a higher force constant (k), what can be said about its vibrational energy levels?

A) They are more closely spaced

B) They are more widely spaced

C) They are less quantized

D) They are lower in energy

22. The angular momentum of a rigid rotator is quantized due to:

A) The constraint of the box

B) The nature of the restoring force

C) The boundary conditions on the angular part of the wave function

D) The quantization of energy

23. What is the physical significance of the wave function (ψ)?

A) It directly represents the position of the particle

B) Its square (|ψ|²) represents the probability density of finding the particle

C) It represents the kinetic energy of the particle

D) It is proportional to the momentum of the particle

24. In the context of the Schrödinger equation, what is an eigenfunction?

A) A function that changes when an operator is applied

B) A function that remains unchanged (except for a multiplicative constant) when an operator is applied

C) A wave function that satisfies boundary conditions

D) A function representing a specific energy level

25. For a particle in a 1D box, the probability of finding the particle is highest at the center of the box for which state?

A) n=1

B) n=2

C) n=3

D) All states

26. The probability density of finding a particle in a given region is proportional to:

A) ψ

B) ∂ψ/∂x

C) ψ²

D) ∫ψ dx

27. The Hamiltonian operator (Ĥ) in the Schrödinger equation represents:

A) The kinetic energy operator

B) The potential energy operator

C) The total energy operator (kinetic + potential)

D) The momentum operator

28. Which of the following is NOT a model system commonly solved using the Schrödinger equation in introductory quantum chemistry?

A) Particle in a box

B) Hydrogen atom

C) Rigid rotator

D) Free particle

29. The zero-point energy of a harmonic oscillator is:

A) Always zero

B) The energy of the ground state

C) The energy of the first excited state

D) Dependent on the amplitude

30. What is the fundamental frequency (ω) of a harmonic oscillator related to?

A) Its mass and wavelength

B) Its mass and force constant

C) Its amplitude and energy

D) Its momentum and position

31. The energy of a vibrational state 'v' for a harmonic oscillator is given by:

A) E_v = vħω

B) E_v = (v+1)ħω

C) E_v = (v+½)ħω

D) E_v = v²ħω

32. What does the quantum number 'v' represent in the harmonic oscillator model?

A) The mass of the oscillator

B) The vibrational energy level

C) The force constant

D) The amplitude of vibration

33. What is the energy of the ground state (v=0) of a quantum harmonic oscillator?

A) 0

B) ħω

C) ½ħω

D) ħω/2π

34. The energy levels of a quantum harmonic oscillator are:

A) Continuously variable

B) Equally spaced

C) Unequally spaced

D) Zero

35. What is the force constant (k) in the harmonic oscillator model related to?

A) The mass of the oscillator

B) The stiffness of the bond or spring

C) The wavelength of vibration

D) The amplitude of oscillation

36. The 'harmonic oscillator' model in quantum mechanics is used to describe:

A) The translational motion of a particle

B) The rotational motion of a molecule

C) The vibrational motion of a bond

D) The electronic transitions in an atom

37. The spacing between adjacent rotational energy levels of a rigid rotator:

A) Is constant

B) Increases with increasing J

C) Decreases with increasing J

D) Is zero

38. What is the degeneracy of the rotational energy level J for a rigid rotator?

A) J

B) 2J

C) 2J+1

D) J+1

39. The energy levels of a rigid rotator are given by the formula:

A) E_J = J(J+1)ħ²/2I

B) E_J = Jħ²/2I

C) E_J = J²ħ²/2I

D) E_J = (J+1)ħ²/2I

40. What is the quantum number that describes the rotational energy levels of a rigid rotator?

A) n

B) l

C) m

D) J

41. For a rigid rotator, the moment of inertia (I) depends on:

A) The charge of the atoms

B) The vibrational frequency

C) The masses of the atoms and the bond length

D) The electronic configuration

42. The 'rigid rotator' model in quantum mechanics is used to describe the rotational motion of:

A) A single atom

B) A diatomic molecule

C) A spherical molecule

D) An electron in an atom

43. What is the wave function for the ground state (n=1) of a particle in a one-dimensional box of length L?

A) √(2/L) sin(πx/L)

B) √(1/L) cos(πx/L)

C) √(2/L) cos(πx/L)

D) √(1/L) sin(2πx/L)

44. If the length of a one-dimensional box is doubled, how does the ground state energy of the particle change?

A) It doubles

B) It quadruples

C) It becomes one-fourth

D) It remains the same

45. What is the ground state energy (lowest possible energy) for a particle in a one-dimensional box of length L?

A) 0

B) h²/8mL²

C) π²h²/2mL²

D) h/2mL

46. What does the quantum number 'n' represent in the 'particle in a box' model?

A) The mass of the particle

B) The length of the box

C) The energy level or state of the particle

D) The charge of the particle

47. In the 'particle in a box' model of length L, what is the quantization condition for the energy levels?

A) E = n²h²/8mL²

B) E = nh/2πmL

C) E = nh²/2mL²

D) E = n²h/2mL

48. For a 'particle in a box' model, what is the boundary condition for the wave function (ψ) at the walls of the box?

A) ψ is maximum

B) ψ is zero

C) ψ is minimum

D) ψ is non-zero and continuous

49. The time-independent Schrödinger equation is used to find:

A) The probability of finding a particle at a certain time

B) The energy levels and wave functions of a stationary state

C) The rate of radioactive decay

D) The momentum of a particle

50. What fundamental equation in quantum mechanics describes the wave function of a quantum system?

A) Heisenberg Uncertainty Principle

B) Schrödinger Wave Equation

C) Bohr Model Equation

D) Pauli Exclusion Principle

Schrödinger wave equation and model systems: particle in a box, rigid rotator and harmonic oscillator - Question Bank