This set of Physical Chemistry Multiple Choice Questions & Answers (MCQs) focuses on “Postulates of Quantum Mechanics”.
1. How is the state of a quantum mechanical system completely specified?
a) By its position in space
b) By its time
c) By its wavefunction
d) By its angular momentum
View Answer
Explanation: A system is completely defined by its wavefunction φ(x,t). This specifies how the particle is moving in space as well as in time. The wavefunction can assume any mathematical form.
2. What is the probability density of a wavefunction φ(x,t) at time t in element of volume dv?
a) φ*(x,t)φ(x,t)dv
b) φ(x,t)dv
c) φ*(x,t)dv
d) φ(x,t)
View Answer
Explanation: For a given wavefunction, the probability density is defined as the product of the complex wavefunction form and the regular wavefunction in a given element of volume. This can also be interpreted as the square of a wavefunction.
3. Is there a quantum mechanical operator for every observable in classical mechanics?
a) Yes, obtained from classical Cartesian coordinates and conjugate momenta
b) Yes, obtained from quantum mechanical coordinates
c) No, only for certain observables
d) No, for no observables at all
View Answer
Explanation: There is a quantum mechanical operator for every observable in classical mechanics. This is obtained from the classical expression for the observable by replacing the conjugate momenta by -iħ\(\frac{\partial}{\partial q}\).
4. By what term should the conjugate momenta be replaced by for conversion of classical observables to quantum mechanical operators?
a) -iħ∇2
b) –\(\frac{\partial}{\partial q}\)
c) -iħ
d) -iħ\(\frac{\partial}{\partial q}\)
View Answer
Explanation: The operator is obtained from the classical expression of the observable in Cartesian coordinates and conjugate momenta by replacing each coordinate q by itself and the conjugate momenta operator by -iħ\(\frac{\partial}{\partial q}\)
5. What are the possible measured values of observable A called?
a) Functions of A
b) Eigenvalues of A
c) Eigenfunctions of A
d) Operators of A
View Answer
Explanation: The possible measured values of A are the eigenvalues of the equation Aφ = aφ. Eigenvalues are the only physically measurable values of an observable acting on a wavefunction.
6. What is the mathematical probability of measuring eigenvalue of an observable for wavefunction φ in a volume of space dv?
a) \(\int_{-V}^V\)φ*φdv
b) \(\int_{-\infty}^\infty\)φdv
c) \(\int_{-\infty}^\infty\)φ*φdv
d) φ×v
View Answer
Explanation: Probability is the square of a given wavefunction over all of space, or the complex conjugate times the regular wavefunction (depending on whether the wavefunction is real or not). The integral bounds must include infinities too encompass all this space.
7. Is it possible for the probability density to ever have a negative value?
a) No, because the square of function is always positive
b) Yes, because the square of a function is always positive
c) No, because the square of a function is not always positive
d) No, because the square of a function is not always positive
View Answer
Explanation: Square of any value or function always yields a positive value. It will be above the x axis in a plot. The manifestation of the square of a wavefunction is its probability density.
8. Is the manifestation of a wavefunction square (to be probability density) a postulate of quantum mechanics?
a) Yes
b) No
c) Depends on the wavefunction form
d) Depends whether the wavefunction is a function of position or time
View Answer
Explanation: Postulate number 4 and 1 of quantum mechanics specify probability density as the square of a wavefunction. This is neither derived from other postulates nor is deduced mathematically from any theory.
9. Can a wavefunction evolve in time?
a) No, it can only evolve in position
b) Yes, it can evolve in time, but not in position
c) No, it can neither evolve in time nor position
d) Yes, it can evolve in time as well as in position
View Answer
Explanation: A wavefunction φ can be a function of both time as well as position. There is no specific form that needs to be followed by this either. The time dependent part of the function involves a phaser and imaginary numbers.
10. The wavefunction of a system changes with time according to the following equation: Hφ(x,t) = iħ\(\frac{\partial \varphi(x,t)}{\partial}\). What does the H term stand for?
a) Hamiltonian of a wavefunction
b) Reduced Plank’s constant
c) Kinetic energy operator
d) Angular momentum operator
View Answer
Explanation: H represents the Hamiltonian operator of a wavefunction. It is the sum of individual kinetic and potential energy operators. The eigenvalues of this operator are the total energy of a quantum mechanical system.
11. The wavefunction of a system changes with time according to the following equation: Hφ(x,t) = iħ\(\frac{\partial \varphi(x,t)}{\partial}\). What does the ħ term stand for?
a) Hamiltonian of a wavefunction
b) Reduced Plank’s constant
c) Kinetic energy operator
d) Angular momentum operator
View Answer
Explanation: ħ is reduced Plank’s constant, that is Plank’s constant divided by 2π. This appears when all classical expressions are replaced by conjugate momenta.
12. The wavefunction of a system changes with time according to the following equation: Hφ(x,t) = iħ\(\frac{\partial \varphi(x,t)}{\partial}\). Can this equation be integrated directly by variable separable?
a) Yes, because φ(x,t) appears on either side with one of them being a differential
b) Yes, because the wavefunction terms can be moved to the right-hand side and integrated
c) No, because φ(x,t) is simply a notation and not the wavefunction form itself
d) No, because φ(x,t) cannot be integrated
View Answer
Explanation: φ(x,t) can assume any function form and so can its corresponding derivative. In the above equation, it should be treated as a function, rather than a variable to be integrated with. Therefore the Schrodinger’s equation in this form cannot be solved easily.
13. What is the Pauli exclusion principle?
a) Two electron wavefunction systems must be antisymmetric to one another
b) Wavefunctions can only exist in superposition of fundamental wavefunctions
c) Electron wavefunctions must be unique to each individual electron
d) The product of uncertainty of two variable pairs cannot exceed h/2π
View Answer
Explanation: Pauli’s exclusion principle states that two electron wavefunction systems must be antisymmetric to one another. Therefore, electrons in the same orbitals must have opposite spin to satisfy the Pauli principle.
14. Are the postulates of quantum mechanics derived from other theories?
a) Yes, they are a special case of classical mechanics
b) Yes, they are derived from thermodynamics and kinetics of atomic motion
c) No, postulates are rules that are not derived
d) Yes, they are the microscopic cases for general relativity
View Answer
Explanation: Postulates are facts assumed to exist without any previous knowledge. For example, the axiom that the sum of angles on a straight line is 180 degrees is a postulate, there is no way this can be proved.
15. Can a classical system’s evolution in time depend upon the observer?
a) Yes, the system will evolve differently if observed
b) Yes, the system’s evolution is purely dependent on the observer
c) No, it will evolve independent of the observer
d) No, it will evolve depending on who the observer is
View Answer
Explanation: A classical system will evolve according to Newton’s laws of motion independent of whether we observe it. This is determined by Newton’s laws of motion that describes trajectory, velocity and acceleration of the particle.
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