This set of Physical Chemistry Multiple Choice Questions & Answers (MCQs) focuses on “Electron Spin”.

1. How was electron spin first detected?

a) Application of magnetic fields showed electrons to rotate differently

b) Orbitals needed electrons of different energy to reside in the same subshell

c) Electron spin was physically detected using a microscope

d) Splitting of certain spectral lines

View Answer

Explanation: An electron can and never will be physically detected since they are clouds rotated around the nucleus at very high speeds. In 1925, Goudsmit proposed electron spin as explains the splitting of certain spectroscopic lines. The relativistic equation for a one electron system shows the existence of electron spin.

2. What is the spin angular momentum analogue in classical mechanics?

a) Rotational inertia

b) Rotational kinetic energy

c) It has no analogue in classical mechanics

d) Moment of inertia

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Explanation: In non-relativistic quantum theory, electron spin must be treated as an additional postulate. It has no analogue in classical mechanics as there is no quantity to replace it. The entire concept of electron spin is quantum mechanical and can only be explained and quantified using quantum mechanical operators.

3. How is the classical Hamiltonian for electron spin written?

a) There is no classical Hamiltonian as there is no analogue for electron spin in classical mechanics

b) It is written the same way as rotational kinetic energy with the mass of an electron

c) It is written the same way as rotational and linear kinetic energy with the mass of an electron

d) It is written the same way as linear angular momentum

View Answer

Explanation: In non-relativistic quantum theory, electron spin must be treated as an additional postulate. It has no analogue in classical mechanics as there is no quantity to replace it. The entire concept of electron spin is quantum mechanical and can only be explained and quantified using quantum mechanical operators. Since there is no classical analogue for electron spin, it cannot have a classical Hamiltonian.

4. What is the value of spin quantum number s?

a) s = ½

b) s = 4πε_{0}

c) s = 1

d) s = 0

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Explanation: The magnitude of electron spin is always designated as s = 1/2. Depending on whether the electron is spin up or spin down, the sign is either positive or negative. This gives definitive answers to spin angular momentum.

5. What is the value of spin angular momentum S?

a) S = 4πε

b) S = \(\frac{\sqrt 3}{2}\)ℏ

c) S = \(\frac{1}{2}\)

d) S = \(\frac{1}{2}\)ℏ^{2}

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Explanation: S = \(\sqrt{s(s+1)}\)ℏ. Since s, the spin quantum number, is always designated as ½, S = \(\sqrt{\frac{1}{2}(\frac{1}{2} + 1})\)ℏ = \(\frac{\sqrt 3}{2}\)ℏ.

6. What is the absolute value of the z-component of spin angular momentum?

a) S_{z} = 4πℏ

b) S_{z} = 0

c) S_{z} = \(\frac{\sqrt 3}{2}\)ℏ

d) S_{z} = \(\frac{\hbar}{2}\)

View Answer

Explanation: The z-component of spin angular momentum is given by S

_{z}= m

_{s}ℏ, where m

_{s}= ±\(\frac{1}{2}\). Hence, the absolute value = S

_{z}= |m

_{s}|ℏ= |±\(\frac{1}{2}\)|ℏ = \(\frac{\hbar}{2}\).

7. What does the operator S^{2} not commute with?

a) Hamiltonian operator

b) L_{z}

c) L^{2}

d) ℏ

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Explanation: Planks reduced constant, ℏ, is not an operator and hence another operator cannot commute with it. S

^{2}commutes with the Hamiltonian, L

^{2}, and L

_{z}such that the energy, spin magnitude, z component of spin, and orbital angular momentum magnitude have simultaneous eigenvalues.

8. Why are two spin functions, α & β used?

a) To describe “spin up” and “spin down” states

b) To describe two electrons in a single orbital

c) To build new standards that describe relatively abstract concepts

d) Electron spin does not involve spatial coordinates, so new functions need to be established to describe them

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Explanation: Spin eigenfunctions do not include spatial coordinates, hence two functions called α and β are used to describe these states. These can help fully mathematically describe respective wavefunctions and eigenfunctions in terms of positional as well as time coordinates.

9. What does the operator S_{z} not commute with?

a) Hamiltonian operator

b) L_{z}

c) L^{2}

d) ℏ

View Answer

Explanation: Planks reduced constant, ℏ, is not an operator and hence another operator cannot commute with it. S

_{z}commutes with the Hamiltonian, L

^{2}, and L

_{z}such that the energy, spin magnitude, z component of spin, and orbital angular momentum magnitude have simultaneous eigenvalues.

10. What is the value of z-component of spin angular momentum for a “spin down” electron?

a) S_{z} = 4πℏ

b) S_{z} = 0

c) S_{z} = \(\frac{1}{2}\)ℏ

d) S_{z} = \(\frac{\hbar}{2}\)

View Answer

Explanation: The z-component of spin angular momentum is given by S

_{z}= m

_{s}ℏ, where m

_{s}= –\(\frac{1}{2}\) for a spin down electron. Hence S

_{z}= m

_{s}ℏ = –\(\frac{1}{2}\)ℏ = –\(\frac{\hbar}{2}\).

11. What is the eigenvalue of operator S^{2} on β?

a) S^{2} = \(\frac{3}{4}\)ℏ^{2}

b) S^{2} = 0

c) S^{2} = \(\frac{\sqrt 3}{2}\)ℏ

d) S^{2} = \(\frac{1}{2}\)ℏ

View Answer

Explanation: S

^{2}= s(s + 1)ℏ

^{2}. Since s, the absolute value of spin quantum number, is always designated as ½, S

^{2}= \(\frac{1}{2}(1 + \frac{1}{2})\)ℏ

^{2}= \(\frac{3}{4}\)ℏ

^{2}. Operating this on a β function will return the same eigenvalue for this operator.

12. What is the eigenvalue of operating S_{z} on α for a “spin up” electron?

a) S_{z} = 4πℏ

b) S_{z} = 0

c) S_{z} = \(\frac{1}{2}\)ℏ

d) S_{z} = \(\frac{\hbar}{2}\)

View Answer

Explanation: The z-component of spin angular momentum is given by S

_{z}= m

_{s}ℏ, where m

_{s}= \(\frac{1}{2}\). Hence, the absolute value = S

_{z}= m

_{s}ℏ = –\(\frac{1}{2}\)ℏ = –\(\frac{\hbar}{2}\). Operating this on the α function will return the same eigenvalue for this operator.

13. What are the possible values of m_{s} given a value of s?

a) m_{s} = -s, +s

b) m_{s} = -s,-s + 1, …., s

c) m_{s} = ±\(\frac{1}{2}\)

d) m_{s} = 0

View Answer

Explanation: The value of magnetic spin is fixed and given by m

_{s}= ±\(\frac{1}{2}\). The sign is positive if the electron is “spin up” and negative if the electron is “spin down”. There is no physical way to determine what is the spin orientation of each electron, but electrons residing in the same orbital must be of opposite spin to conserve energy and angular momentum.

14. What is the value of the spin eigenfunction integral ∫ α^{*} αdτ?

a) ∫ α^{*} αdτ = 1

b) ∫ α^{*} αdτ = 0

c) ∫ α^{*} αdτ = \(\frac{3}{4}\)ℏ^{2}

d) ∫ α^{*} αdτ = \(\frac{1}{2}\)ℏ

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Explanation: The α & β function is normalized over all space with respect to its complex conjugate. Spin eigenfunctions do not include spatial coordinates, hence two functions called α and β are used to describe these states.

15. What is the value of the spin eigenfunction integral ∫ α^{*} βdτ?

a) ∫ α^{*} αdτ = 1

b) ∫ α^{*} αdτ = 0

c) ∫ α^{*} αdτ = \(\frac{3}{4}\)ℏ^{2}

d) ∫ α^{*} αdτ = \(\frac{1}{2}\)ℏ

View Answer

Explanation: The α & β function is normalized over all space with respect to its complex conjugate. They are also orthogonal to each other. Spin eigenfunctions do not include spatial coordinates, hence two functions called α and β are used to describe these states.

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