# Physical Chemistry Questions and Answers – Quantum Theory – Expectation Values and Superposition

This set of Physical Chemistry Multiple Choice Questions & Answers (MCQs) focuses on “Quantum Theory – Expectation Values and Superposition”.

1. The possible outcomes for measurement of a four-sided dice are x = 1,2,3,4. What is the expectation outcome (mean)?
a) 1.5
b) 2.5
c) 1
d) 2

Explanation: Mathematical expectation is defined as $$\Sigma_{i=1}^n\frac{x_i}{n}$$. The expectation is thus $$\frac{1+2+3+4}{4}$$ = 2.5

2. The possible outcomes for measurement of a four-sided dice are x = 1,2,3,4. What is the variance of possible outcomes?
a) 1.5
b) 1.1
c) 1.25
d) 1.75

Explanation: Variance is the deviation from the mean and is defined by $$\Sigma_{i=1}^n(\frac{x_i-x}{n})^2$$ = $$\frac{1}{4}$$((2.5 – 1)2 + (2.5 – 2)2 + (2.5 – 3)2 + (2.5-4)2) = $$\frac{2.5}{2}$$ = 1.25

3. The possible outcomes for measurement of a four-sided dice are x = 1,2,3,4. What is the standard deviation of possible outcomes?
a) 1.5
b) 1.1
c) 1.25
d) 1.75

Explanation: Standard deviation is defined as the square root of variance = $$\sqrt{\Sigma_{i=1}^n(\frac{x_i-x}{n})^2} = \sqrt\frac{2.5}{2} = \sqrt{1.25} \approx 1.1$$

4. If there are several superimposed normalized wavefunctions Φn, what is the sum of all the wavefunction coefficients squared cn2?
a) 0
b) -1
c) ∞
d) 1

Explanation: For a normalized wavefunction, 1 = ∫ φ*φ = ∫ (ΣncnΦn)*mcmΦm)dτ = Σn$$c_n^*c_n$$ ∫ $$\Phi_n^*\Phi_n$$dτ = Σn|cn|2 = 1. Since all the wavefunctions are normalized, this is always true.

5. What is the average <x> of a system when an observable A has a continuous set of eigenvalues?
a) <x> = ∫ ψ*(x)xψ(x) dx
b) <x> = ∫ ψ*(x)ψ(x) dx
c) <x> = ∫ ψ(x) dx
d) <x> = ∫ ψ*(x) dx

Explanation: The average is the physical quantity times the probability density, which is ∫ ψ*(x)ψ(x) dx. This gives the average value in a specified one-dimensional coordinate system. ∫ ψ*(x)ψ(x) dx is the probability density. This applies for any quantum mechanical system.

6. A particle is in the quantum state ψ = n1Φ1 + n2Φ2, that is the superposition of 2 eigenfunctions of energy Φ1 and Φ2 with energy eigenvalues E1 and E2. What is the probability of measuring E2?
a) n1
b) |n1|2
c) |n2|2
d) n2

Explanation: Since ψ is normalized, and Φ1 and Φ2 are orthogonal, |n1|2 + |n2|2 = 1. This means that the probability of measuring E2 is |n2|2 as the total probability must add to 1. Individual eigenstates are always orthogonal to each other.

7. A particle is in the quantum state ψ = n1Φ1 + n2Φ2, that is the superposition of 2 eigenfunctions of energy Φ1 and Φ2 with energy eigenvalues E1 and E2. What is the average energy for the given quantum state?
a) <E> = |n1|2E1 + |n2|2E2
b) <E> = |n1|2 + |n2|2
c) <E> = $$\frac{E1 + E2}{2}$$
d) <E> = E1 + E2

Explanation: Average energy is the weighted average of individual superimposed wavefunction energies. Since each energy has a probability of |n1|2 and |n2|2 respectively, multiplying each by their energies gives an average energy value for the given superposition state.

8. What is a standing wave?
a) A wave, formed from the superposition of two other waves, that oscillates in space but not time
b) A wave that oscillates in time but does not change peak amplitude
c) A wave that is essentially stationary and performs no oscillations in space or time
d) A wave that appears to move in 2-D space, but is stationary when viewed in 3-D

Explanation: In classical mechanics, a standing wave has a fixed peak amplitude, but oscillates in time. This can occur when two waves moving in opposite directions interfere or when a medium is moving in the opposite direction to the wave.

9. A particle is in the quantum state ψ = n1Φ1 + n2Φ2, that is the superposition of 2 eigenfunctions of energy Φ1 and Φ2 with energy eigenvalues E1 and E2. What is the standard deviation in energy for the given quantum state?
a) σE = ((|n1|2E1 + |n2|2E2) – (|n1|2E1 + |n2|2E2)2)1/2
b) σE = ((E1 + E2) – (E1 + E2)2)1/2
c) σE = ((|n1|2E1 + |n2|2E2) – (|n1|2E1 + |n2|2E2)2)1/2 + ((|n1|2E1 + |n2|2E2))1/2
d) σE = (|n1|2E1 + |n2|2E2)2

Explanation: Standard deviation σE = [<E2> – <E>2]2; <E2> = ∫ Φ*H2Φdτ = |n1|2E12 + |n2|2E22 ; σE = ((|n1|2E1 + |n2|2E2) – |n1|2E1 + |n2|2E2)2)1/2

10. What is the meaning of variance of a population sample?
a) Expected value of the sample
b) Maximum occurrences of a data point in that sample
c) Point at midpoint of frequency distribution

Explanation: Variance is the deviation of the data points from the mean. The higher this deviation, the higher the variance. Expected value is the average (mean), maximum occurrences in the mode, and point at the midpoint of a frequency distribution is median.

11. What is the mathematical representation of variance σx2?
a) σx2 = < (x – < x >) >2
b) σx2 = (x – < x >)
c) σx2 = (< x >)2
d) σx2 = (x)2

Explanation: Variance is the deviation of data points from the mean. It is the sum of the square of differences between each data point and the average; it is the square of standard deviation.

Sanfoundry Global Education & Learning Series – Physical Chemistry.

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