This set of Physical Chemistry Multiple Choice Questions & Answers (MCQs) focuses on “Time Dependent Schrodinger Equation”.

1. What is one shortcoming of separating variables and solving the time dependent Schrodinger equation?

a) The wavefunction is a function of position only

b) Potential energy in the Hamiltonian can be written as a function of position only

c) Complex wavefunction forms cannot be solved for

d) Kinetic energy in the Hamiltonian can be written as a function of position only

View Answer

Explanation: The kinetic energy term is a function of position only because of the Laplacian, but this is not a shortcoming. Potential energy must be a function of position only to isolate variables and write a time dependent Schrodinger equation. This is a conservative system.

2. What is the initial wavefunction condition?

a) φ(x,0) = φ(x)

b) φ(x,0) = φ(x,t)

c) φ(x) = φ(0)

d) φ(x,t) = φ(x,0)

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Explanation: Initially, the wavefunction has no dependence on time, hence it is only a function of position at time t = 0. This only applies as the initial boundary condition for solving the time dependence part of the Schrodinger equation.

3. What is the expression for time dependence f(t) for a conservative system?

a) f(t) = A sin t

b) f(t) = A sin t + B cos t

c) f(t) = e^{-it}

d) f(t) = \(e^{-\frac{iEt}{\hbar}}\)

View Answer

Explanation: Separating the Schrodinger equation into two differential equations, we get the second as –\(\frac{\hbar}{i} \frac{df(t)}{dt}\) = Ef(t) —→ solving this with the boundary condition φ(x,0) = φ(x), gives f(t) = \(e^{-\frac{iEt}{\hbar}}\).

4. What is the time dependent wavefunction for a conservative one-dimensional system in an eigenstate of H?

a) φ(x,t) = φ(x)[A sin t + B cos t]

b) φ(x,t) = φ(x).t

c) φ(x,t) = φ(x)e^{-it}

d) φ(x,t) = φ(x)\(e^{-\frac{iEt}{\hbar}}\)

View Answer

Explanation: Separating the Schrodinger equation into two differential equations, we get the second as –\(\frac{\hbar}{i} \frac{df(t)}{dt}\) = Ef(t) —→ solving this with the boundary condition φ(x,0) = φ(x), gives f(t) = \(e^{-\frac{iEt}{\hbar}}\). Inserting this into the Schrodinger equation gives φ(x,t) = φ(x)\(e^{-\frac{iEt}{\hbar}}\).

5. If the wavefunction φ(x,t) is complex, is the probability density real?

a) Yes, because probability density is a square and hence cannot be complex

b) Yes, because probability density doesn’t depend on the wavefunction

c) No, because a complex wavefunctions yields a complex probability density

d) Depends on the wavefunction form

View Answer

Explanation: φ

^{*}(x,t)φ(x,t) = [φ

^{*}(x)\(e^{\frac{iEt}{\hbar}}\)][φ(x)\(e^{-\frac{iEt}{\hbar}}\)] = φ

^{*}(x)φ(x). The complex part of the wavefunction thus disappears and the probability density depends on position only. This system is referred to as being in stationary state.

6. What is the simplified mathematical expression for φ^{*}(x,t)φ(x,t) for a one-dimensional wavefunction conservative system?

a) φ(x)\(e^{-\frac{2iEt}{\hbar}}\)

b) φ^{*}(x)φ(x)

c) φ^{2}(x)\(e^{-\frac{iEt}{\hbar}}\)

d) φ^{2}(x)\(e^{-\frac{2iEt}{\hbar}}\)

View Answer

Explanation: φ

^{*}(x,t)φ(x,t) = [φ

^{*}(x)\(e^{\frac{iEt}{\hbar}}\)][φ(x)\(e^{-\frac{iEt}{\hbar}}\)] = φ

^{*}(x)φ(x). The time, as well as the imaginary component of the wavefunction disappears through this mathematical manipulation. This assumed that the time component can be represented through Euler’s simplified phasor relation.

7. What does it mean for a quantum mechanical system to be in stationary state?

a) Eigenvalues are real numbers

b) Eigenvalues are complex numbers

c) Probability density is independent of time

d) The atoms are not moving relative to each other

View Answer

Explanation: For a mechanical system to be in a stationary state, the probability density must be a function of position only. There should be no time component involved. Moreover, there should be only one special component rather than multiple (for e.g. only x, rather than x, y, and z).

8. What is it called when the probability density of a system is independent of time?

a) System is in a stationary state

b) System has no real eigenvalues

c) System has no imaginary eigenvalues

d) System can be treated classically

View Answer

Explanation: For a system to be independent of time, it must depend on position solely. The wavefunction only depends on special components such as x, y, and z. This is called stationary state because the quantum mechanical system is “stationary” with change in time. Hence, the wavefunction can be represented as φ(x).

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