Okay, so I need to figure out the oscillation frequency of the z-component of the particle's magnetic moment after the magnetic fields change. Let me start by recalling what I know about spin-1/2 particles and their behavior in magnetic fields.

First, the particle is a spin-1/2, which means its magnetic moment μ is related to its spin S_z by μ = γ*S_z. Here, γ is the gyromagnetic ratio. The initial state is that the particle is aligned with the magnetic field pointing in the +Z direction. So initially, the spin is in the +Z direction, which means S_z is at its maximum value, ħ/2. Therefore, the initial magnetic moment μ is γ*(ħ/2). But wait, maybe the units are such that ħ is absorbed into γ, so maybe μ is just γ times S_z. I think that's how it's usually presented.

Now, the magnetic field is suddenly turned off, and another field is turned on along the +Y direction with the same magnitude B. So the magnetic field after the change is along Y, which is perpendicular to the original Z direction. The problem says that the average of the z-component of the magnetic moment starts oscillating. I need to find its oscillation frequency.

Hmm, okay. Let me think about how the spin state evolves when the magnetic field changes. The magnetic field applies a torque on the magnetic moment, causing the spin to precess. The torque is given by τ = μ × B. For a spin-1/2 particle, the equation of motion is dS/dt = γ B × S. Wait, actually, the torque is τ = dL/dt = d(μ)/dt, but since μ is proportional to S, it's μ = γ S, so dμ/dt = γ dS/dt. But torque is also τ = μ × B, so combining these gives γ dS/dt = γ S × B. Therefore, dS/dt = (1/γ) (S × B). Wait, maybe I should check the units here. Alternatively, the equation of motion for spin in a magnetic field is dS/dt = γ B × S. Because the torque τ = dL/dt = d(μ)/dt = γ dS/dt, and torque is also τ = μ × B. So γ dS/dt = (γ S) × B → dS/dt = (1/γ)(γ S × B) = S × B. Wait, that can't be right because the units would be different. Maybe I messed up the factors. Let's think again.

The torque is τ = dL/dt = d(μ)/dt. But μ = γ S, so dμ/dt = γ dS/dt. The torque is also τ = μ × B. Therefore, γ dS/dt = γ S × B. Dividing both sides by γ, we get dS/dt = S × B. Hmm, but the units here would be S has units of angular momentum (kg·m²/s), B is tesla, so S × B would have units kg·m²/s · T. Wait, torque is in N·m, which is kg·m²/s². Hmm, maybe there's a factor missing. Alternatively, perhaps the correct equation is dS/dt = (γ/2) B × S. Wait, maybe I need to recall the correct expression. Let me think. The gyromagnetic ratio γ is defined such that the precession frequency is ω = γ B. For a spin-1/2 particle, the magnetic moment is μ = γ S. The torque is τ = μ × B = γ S × B. The angular momentum equation is dS/dt = τ / γ. Wait, that would be dS/dt = (γ S × B)/γ = S × B. So yeah, dS/dt = S × B. But that seems to have units of angular momentum per second (since S is angular momentum, B is tesla; S × B would have units kg·m²/s · T. Wait, T is kg/(s·A), so kg·m²/s * kg/(s·A) = kg²·m²/(s²·A). Hmm, not sure. Maybe I should focus on the dynamics rather than units for now.

Alternatively, the equation is dS/dt = (γ/2) B × S. Wait, maybe the correct equation includes a factor of 1/2. Let me check. For a spin-1/2 particle, the magnetic moment is μ = γ S, but the spin precession frequency is ω = γ B. Wait, for a magnetic moment in a magnetic field, the torque is μ × B, so dS/dt = (μ × B)/γ. Since μ = γ S, then dS/dt = (γ S × B)/γ = S × B. So the equation is dS/dt = S × B. But the standard precession equation is dS/dt = ω × S, where ω is the angular frequency. So in this case, ω would be B. Wait, but the units don't seem to match because ω should be in radians per second, and B is in tesla. Hmm, maybe there's a factor missing. Alternatively, perhaps the equation should be dS/dt = (γ B) × S. Then the precession frequency would be γ B. Wait, that makes more sense dimensionally. Let's see. The torque is τ = μ × B = γ S × B. Then dS/dt = τ / γ = (γ S × B)/γ = S × B. So that would give dS/dt = S × B. But then the equation is S × B, which would have units of angular momentum per second. Alternatively, if we have dS/dt = γ B × S, then the precession frequency is γ B. Let me check. Suppose S is along z, then dS/dt = γ B × S. If S is along z, then the cross product would be in the x or y direction. Wait, if B is along y, then γ B × S, with S along z, would be in the x direction. So the spin would start precessing around the y-axis. The precession frequency would be γ B. Hmm, that seems right. So the equation should be dS/dt = γ B × S. But earlier, when I derived it from torque and angular momentum, I got dS/dt = S × B. So there's a discrepancy here. Maybe I made a mistake in the derivation. Let me go through it again.

Torque τ = dL/dt = d(μ)/dt. But μ = γ S, so dμ/dt = γ dS/dt. Torque τ is also equal to μ × B. Therefore, γ dS/dt = (γ S) × B. Dividing both sides by γ, we get dS/dt = S × B. So that's correct. But then, if the equation is dS/dt = S × B, then the precession frequency would be related to B. Let me see. If S is pointing along the z-axis, then dS/dt = (S × B), where B is along y. The cross product of z and y is x. So the derivative of S (which is along z) is along x. The magnitude of dS/dt would be |S||B| sin(90°) = S B. So the rate of change of S is S B. But angular momentum changes at a rate proportional to the component perpendicular to the spin direction. Wait, maybe another approach. Let's consider the equation dS/dt = ω × S, where ω is the angular frequency. Then, comparing with dS/dt = S × B, we can write ω × S = S × B. So ω must be along B, but that would imply ω is perpendicular to S. Wait, that doesn't make sense. Alternatively, maybe I need to think in terms of angular frequency. Let me consider a spin precessing around the y-axis with angular frequency ω. Then the equation would be dS/dt = ω × S. But according to our previous equation, dS/dt = S × B. So equating these, ω × S = S × B. So ω must be equal to B, but that's only possible if B is along the same direction as ω, which is along B. Wait, but B is along y. So ω would have to be along y. Then ω × S would be along x if S is along z. But S × B would be along x as well. So perhaps ω is equal to B. Wait, but that would mean the precession frequency is B. But in reality, the precession frequency is γ B. So maybe there's a factor of γ missing in the equation.

Alternatively, perhaps the correct equation is dS/dt = γ B × S. Then, the precession frequency would be γ B. Let's see. If that's the case, then the solution would involve the Larmor frequency, which is γ B. So the oscillation frequency would be γ B. But looking at the choices, option A is gamma*B, which is option A. But the problem states that the magnetic field is turned on along +Y. So the precession would be around Y, causing the spin component in the original Z direction to oscillate. Let's model this.

The initial state is along +Z. When the magnetic field is turned on along +Y, the Hamiltonian becomes H = -μ·B = -γ B S_y. Wait, because the magnetic moment is μ = γ S_z, so the Hamiltonian would be H = -μ·B = -γ B S_z if B is along z, but in this case, B is along Y. Wait, no. Wait, the magnetic moment is μ = γ S_z. Wait, no, the problem states that mu is equal to gamma*S_z. Wait, that's a bit confusing. Let me check the problem statement again: "There is a spin 1/2 particle with magnetic moment mu (= gamma*Sz), aligned with a magnetic field pointing in the +Z direction."

So the magnetic moment is μ = γ S_z. So the Hamiltonian when the magnetic field is along +Z is H = -μ·B = -γ B S_z. But after the field is turned off and turned on along +Y, the Hamiltonian becomes H = -μ·B, but B is along Y. Wait, but the magnetic field is turned off in the original direction and another is turned on along +Y. So after the change, the magnetic field is along +Y, so H = -μ·B = -γ B S_y. Wait, because B is along Y, so B = B \hat{y}, so μ·B = μ B_y = γ S_z B. Wait, no. Wait, if B is along Y, then S_y is the component of spin along Y. Wait, but μ is along Z, so μ·B would be μ B_y, but if B is along Y, then B_y = B, so H = -γ B S_z. Wait, that can't be. Wait, maybe I'm getting confused here.

Wait, let's clarify. The magnetic moment is μ = γ S_z. So if the magnetic field is along Y, then the magnetic moment is still μ = γ S_z. But the energy is H = -μ·B. So if B is along Y, then μ·B = γ S_z * B. So H = -γ B S_z. Wait, that's the same as when the field was along Z. That can't be right. Wait, maybe I misunderstood the problem. Let me read it again.

The problem says: "Suddenly, the magnetic field is turned off, and another one is turned on along the +Y direction; both fields have the same magnitude, B." So the initial field is along Z, then it's turned off, then another field along Y with same magnitude B. So the initial Hamiltonian was H_initial = -γ B S_z. Then the field is turned off, and a new field along Y is applied. So the new Hamiltonian is H_final = -γ B S_y. Because the magnetic moment is μ = γ S_z, so μ·B = γ S_z * B_y. If B is along Y, then B_y = B, and B_z = 0. So H_final = -γ B S_z. Wait, that's the same as before. That can't be. Wait, no, because if the magnetic field is along Y, then the operator is S_y. But the magnetic moment is μ = γ S_z. So the dot product μ·B would be γ S_z * B_y. But if B is along Y, then B_y = B, and B_z = 0. So H_final = -γ B S_z. Wait, but that's the same as before. That seems odd. Wait, maybe I'm making a mistake here. Let me think again.

Wait, perhaps the magnetic moment is μ = γ S_z, so it's along Z. When the magnetic field is turned on along Y, the energy is H = -μ·B. If B is along Y, then μ·B = γ S_z * B. Wait, no. Wait, if B is along Y, then B has components (0, B, 0). So μ·B = (γ S_z) * B_y + (0) * S_x + (0) * S_z? Wait, no. Wait, μ is a vector, right? So μ = (0, 0, γ S_z). Then, if B is along Y, B = (0, B, 0). So μ·B = 0*0 + 0*B + γ S_z * 0 = 0. Wait, that can't be right. That would imply that the Hamiltonian is zero, which doesn't make sense. Hmm, maybe I misunderstood the problem. Let me read again.

Wait, the problem states: "There is a spin 1/2 particle with magnetic moment mu (= gamma*S_z), aligned with a magnetic field pointing in the +Z direction." So the magnetic moment is along Z. Then, the magnetic field is turned off, and another is turned on along +Y with the same magnitude B. So the initial state is along +Z, then the magnetic field is turned off (so no B), then another field is applied along +Y. Wait, but when the field is turned on, the energy becomes H = -μ·B. If B is along Y, and μ is along Z, then the dot product is zero. That would mean the Hamiltonian is zero, which would mean no energy. That can't be. So perhaps I'm misunderstanding the problem. Maybe the magnetic moment is not along Z, but the spin is along Z. Wait, the problem says "magnetic moment mu (= gamma*S_z)", so the magnetic moment is proportional to S_z. So if the spin is along Z, then the magnetic moment is along Z. Then, when the magnetic field is turned on along Y, the Hamiltonian would involve the spin operator along Y. But if the magnetic moment is along Z, then the dot product with B along Y would be zero. That can't be. So maybe there's a mistake here. Wait, maybe the magnetic moment is along Z, but when the field is turned on along Y, the Hamiltonian is H = -μ·B, which would involve the component of B along the direction of the magnetic moment. Wait, but if B is along Y, and the magnetic moment is along Z, then μ·B = 0. So H = 0. That doesn't make sense. The problem must have intended that the magnetic moment is along the direction of the magnetic field. So if the magnetic field is along Y, then the magnetic moment is along Y. But the problem says mu = gamma*S_z. So S_z is the spin component along Z. Wait, maybe the problem is that the spin is initially along Z, so the magnetic moment is along Z. Then, after the field is turned on along Y, the magnetic moment is now aligned with the new field. Wait, but the magnetic moment is fixed as mu = gamma*S_z. Wait, no, the spin can precess, so the direction of the spin can change. Wait, but the magnetic moment is proportional to the spin. So if the spin precesses, the magnetic moment vector also precesses. So the initial state is spin along Z, so magnetic moment along Z. Then, when the field is turned on along Y, the magnetic moment would now feel a torque causing it to precess around Y. So the magnetic moment would start moving in a circle around the Y-axis. But since the magnetic moment is proportional to the spin, the spin would precess as well. The problem states that the average of the z-component of the magnetic moment starts oscillating. So even though the magnetic moment is now in a state that's precessing around Y, the z-component would oscillate in time. Let me model this.

The initial state is |+z>, so the spin is along Z. The magnetic moment is mu = gamma S_z. When the field is turned on along Y, the Hamiltonian becomes H = -mu·B = -gamma B S_y. Because B is along Y, the Hamiltonian is proportional to S_y. The time evolution of the spin state in this Hamiltonian will cause the spin to precess around the Y-axis. The energy eigenstates for this Hamiltonian are states with definite S_y, but the initial state |+z> is not an eigenstate of S_y. So we need to express |+z> in terms of the eigenstates of S_y. The eigenstates of S_y are |+y> and |-y>, which are superpositions of |+z> and |-z>. Specifically, |+y> = (|+z> + |-z>)/sqrt(2), and |-y> = (|+z> - |-z>)/sqrt(2).

The time evolution under H = -gamma B S_y would cause the state to oscillate between these states. The energy eigenvalues are E_+ = -gamma B (1/2), since S_y has eigenvalues +/- 1/2. Wait, the spin-1/2 operator S_y has eigenvalues +1/2 and -1/2 in units where ħ is absorbed. Wait, actually, S_y has eigenvalues ±ħ/2, but if we're working in units where ħ=1, then S_y has eigenvalues ±1/2. So the energy eigenvalues E_+ = -gamma B (1/2), E_- = gamma B (1/2). The energy difference between the two states is E_+ - E_- = -gamma B. The time evolution would cause transitions between the states. The initial state |+z> can be written as (|+y> + |-y>)/sqrt(2). So under time evolution, each eigenstate picks up a phase factor. The state at time t would be |psi(t)> = (e^{-i E_+ t} |+y> + e^{-i E_- t} |-y>)/sqrt(2). Substituting the eigenvalues, E_+ = -gamma B (1/2), E_- = gamma B (1/2). So the state becomes [e^{i (gamma B/2) t} |+y> + e^{-i (gamma B/2) t} |-y> ] / sqrt(2). Now, let's express this back in terms of |+z> and |-z>. Recall that |+y> = (|+z> + |-z>)/sqrt(2), |-y> = (|+z> - |-z>)/sqrt(2). Substituting these into the state:

|psi(t)> = [ e^{i (gamma B/2) t} (|+z> + |-z>)/sqrt(2) + e^{-i (gamma B/2) t} (|+z> - |-z>)/sqrt(2) ] / sqrt(2)

Combine the terms:

= [ (e^{i (gamma B/2) t} + e^{-i (gamma B/2) t}) |+z> + (e^{i (gamma B/2) t} - e^{-i (gamma B/2) t}) |-z> ] / (sqrt(2) * sqrt(2))

Simplify the denominators: sqrt(2)*sqrt(2) = 2.

So,

|psi(t)> = [ (2 cos(gamma B t / 2)) |+z> + (2i sin(gamma B t / 2)) |-z> ] / 2

Which simplifies to:

|psi(t)> = cos(gamma B t / 2) |+z> + i sin(gamma B t / 2) |-z>

Now, the expectation value of the z-component of the magnetic moment is mu_z = gamma <S_z>. Since mu = gamma S_z, the expectation value of mu_z is gamma times the expectation value of S_z. Let's compute <S_z>:

<S_z> = <psi(t)| S_z |psi(t)>

Expressing |psi(t)> as a linear combination of |+z> and |-z>:

|psi(t)> = cos(theta) |+z> + i sin(theta) |-z>, where theta = gamma B t / 2.

So,

<S_z> = [cos(theta) |+z> + i sin(theta) |-z> ]^† S_z [cos(theta) |+z> + i sin(theta) |-z> ]

Since S_z |+z> = (1/2) |+z>, and S_z |-z> = (-1/2) |-z>,

So,

<S_z> = [cos(theta) (1/2) + (-i sin(theta)) (-1/2) ] = (1/2)(cos(theta) + i sin(theta)) + (1/2)(i sin(theta) + cos(theta)) )

Wait, let me compute this step by step. The bra vector is [cos(theta), i sin(theta)] and the ket vector is [cos(theta), i sin(theta)]^T. So,

<S_z> = cos(theta)(1/2) cos(theta) + cos(theta)(-i/2) i sin(theta) + i sin(theta)(1/2) cos(theta) + i sin(theta)(-i/2) i sin(theta)

Wait, maybe it's easier to compute it as:

<S_z> = |cos(theta)|^2 (1/2) + |i sin(theta)|^2 (-1/2) + cos(theta)(i sin(theta)) (0) + cos(theta)(i sin(theta)) (0)

Wait, no, actually, the matrix elements come from the outer product. Let me think again.

The state |psi(t)> is written as a column vector:

[ cos(theta)
  i sin(theta) ]

The adjoint is [ cos(theta), -i sin(theta) ]

Then, <S_z> is the sum over the diagonal elements of the matrix representation. For a spin-1/2 system, S_z in matrix form is (1/2) [[1, 0], [0, -1]]. So,

<S_z> = [cos(theta), -i sin(theta)] * [[1/2, 0], [0, -1/2]] * [cos(theta), i sin(theta)]^T

Calculating this:

First, multiply the matrix with the ket vector:

[[1/2, 0], [0, -1/2]] * [cos(theta), i sin(theta)]^T = [ (1/2) cos(theta), (-1/2) i sin(theta) ]

Then multiply by the bra vector [cos(theta), -i sin(theta)]:

(1/2 cos(theta)) cos(theta) + (-1/2 i sin(theta)) (-i sin(theta))

= (1/2) cos²(theta) + (-1/2)(i * -i) sin²(theta)

Note that i * -i = -i² = -(-1) = 1.

So,

= (1/2) cos²(theta) + (1/2) sin²(theta)

= (1/2)(cos²(theta) + sin²(theta)) = 1/2.

Wait, that's interesting. The expectation value of S_z is 1/2 at all times? That can't be right. If the state is a superposition of |+z> and |-z>, then the expectation value of S_z should oscillate. But according to this calculation, it's always 1/2. That seems contradictory. Let me check the calculation again.

Wait, perhaps I made a mistake in the matrix multiplication. Let me recompute:

The matrix [[1/2, 0], [0, -1/2]] multiplied by the ket vector [cos(theta), i sin(theta)] is:

First component: (1/2) cos(theta) + 0 * i sin(theta) = (1/2) cos(theta)

Second component: 0 * cos(theta) + (-1/2)(i sin(theta)) = (-i/2) sin(theta)

Then, the bra vector [cos(theta), -i sin(theta)] multiplied by this result is:

cos(theta) * (1/2 cos(theta)) + (-i sin(theta)) * (-i/2 sin(theta))

= (1/2) cos²(theta) + (-i)(-i)/2 sin²(theta)

= (1/2) cos²(theta) + (i²)/2 sin²(theta)

But i² = -1, so:

= (1/2) cos²(theta) - (1/2) sin²(theta)

Therefore, <S_z> = (1/2)(cos²(theta) - sin²(theta)) = (1/2) cos(2 theta)

Ah, there we go. So the expectation value of S_z is (1/2) cos(2 theta), where theta = gamma B t / 2. Therefore, substituting theta:

<S_z> = (1/2) cos(2 * gamma B t / 2) = (1/2) cos(gamma B t)

Therefore, the expectation value of the z-component of the magnetic moment is mu_z = gamma <S_z> = gamma * (1/2) cos(gamma B t) = (gamma B / 2) cos(gamma B t)

Wait, but the problem states that the average of the z-component of the particle’s moment starts oscillating. The average would be the expectation value, which we just found is (gamma B / 2) cos(gamma B t). The oscillation frequency is the coefficient of t inside the cosine, which is gamma B. However, the amplitude is (gamma B / 2), but the frequency is gamma B.

Wait, but the options are:

A. gamma*B

B. sqrt(2)*gamma*B

C. gamma*B/sqrt(2)

D. gamma*B/2

But according to this calculation, the average oscillates at frequency gamma B. However, the options have A as gamma*B, which would be the frequency. But wait, the oscillation is cos(gamma B t), which has frequency gamma B. So the oscillation frequency is gamma B. But I need to confirm this.

Alternatively, perhaps I made a mistake in the calculation. Let me check again.

Wait, the expectation value of S_z was found to be (1/2) cos(gamma B t). Therefore, the magnetic moment is mu_z = gamma * (1/2) cos(gamma B t) = (gamma B / 2) cos(gamma B t). The cosine function has an argument of gamma B t, so the frequency is gamma B. Therefore, the oscillation frequency is gamma B, which is option A. However, the problem states that the average starts oscillating. The average would be the expectation value, which is (gamma B / 2) cos(gamma B t). So the oscillation frequency is gamma B, which matches option A. But wait, the problem asks for the oscillation frequency, which is the coefficient of t in the cosine, which is gamma B. So the answer should be A.

But wait, let me think again. The torque causes the spin to precess around the Y-axis. The precession frequency for spin-1/2 particles in a magnetic field is given by the Larmor frequency, which is omega = gamma B. But in this case, the field is along Y, so the precession is around Y. However, the question is about the oscillation of the z-component of the magnetic moment. The z-component oscillates due to the precession, and the frequency of that oscillation is indeed the Larmor frequency, gamma B. Therefore, the correct answer should be A.

Wait, but some sources mention that the oscillation frequency of the expectation value is actually half the Larmor frequency. For example, in the case of spin coherent oscillations, the transition frequency (like in ESR) is gamma B / 2. Wait, but in this case, the expectation value of S_z is (1/2) cos(gamma B t), so the frequency is gamma B. Therefore, the oscillation frequency is gamma B. Therefore, option A is correct. But let me check again.

Alternatively, perhaps there's a factor of 1/2 due to the spin being 1/2. Let me think. The energy levels are E = ± gamma B (1/2). The energy difference is gamma B. The frequency is energy difference divided by h-bar, but since we're working in units where ħ=1, the frequency is gamma B. Therefore, the oscillation frequency is gamma B. So the answer is A.

Wait, but I recall that in the two-level system with Rabi oscillations, the transition frequency is indeed the energy difference, which in this case would be gamma B. Therefore, the oscillation frequency is gamma B. So the correct answer should be A.
</think>

To determine the oscillation frequency of the z-component of the particle's magnetic moment after the magnetic field is turned on along the +Y direction, we analyze the system's dynamics.

1. **Initial State and Magnetic Field Change**: The particle starts in a spin-1/2 state aligned with the +Z direction. The magnetic moment is given as $\mu = \gamma S_z$, where $\gamma$ is the gyromagnetic ratio, and $S_z$ is the spin operator in the z-direction. Initially, the magnetic field is along +Z, so the spin is in the +Z eigenstate.

2. **Magnetic Field Switching**: The magnetic field is suddenly turned off along the +Z direction and replaced with a field along the +Y direction of the same magnitude $B$. The Hamiltonian for the system after the field is turned on becomes $H = -\mu \cdot B = -\gamma B S_y$, where $S_y$ is the spin operator in the y-direction.

3. **Time Evolution and Oscillation**: The eigenstates of $S_y$ are the spin-up and spin-down states along the y-axis, denoted as $|+y\rangle$ and $|-y\rangle$, with eigenvalues $\pm 1/2$. The initial state $|+z\rangle$ can be expressed as a superposition of these eigenstates:
   \[
   |+z\rangle = \frac{1}{\sqrt{2}} \left( |+y\rangle + |-y\rangle \right).
   \]
   Under the Hamiltonian $H = -\gamma B S_y$, the time evolution of the state introduces a phase difference between the $|+y\rangle$ and $|-y\rangle$ components. The state at time $t$ becomes:
   \[
   |\psi(t)\rangle = \cos\left(\frac{\gamma B t}{2}\right) |+z\rangle + i \sin\left(\frac{\gamma B t}{2}\right) |-z\rangle.
   \]

4. **Expectation Value of $S_z$**: The expectation value of $S_z$ is calculated as:
   \[
   \langle S_z \rangle = \langle \psi(t) | S_z | \psi(t) \rangle = \frac{1}{2} \cos(\gamma B t).
   \]
   The oscillation frequency of $\langle S_z \rangle$ is determined by the argument of the cosine function, which is $\gamma B t$. Thus, the oscillation frequency is $\gamma B$.

5. **Magnetic Moment Oscillation**: Since the magnetic moment $\mu_z = \gamma S_z$, the oscillation frequency of $\mu_z$ is the same as that of $\langle S_z \rangle$. Therefore, the oscillation frequency of the z-component of the magnetic moment is $\gamma B$.

6. **Answer Verification**: The problem's options are:
   - A. $\gamma B$
   - B. $\sqrt{2} \gamma B$
   - C. $\gamma B / \sqrt{2}$
   - D. $\gamma B / 2$
   
   From the analysis, the oscillation frequency is $\gamma B$, which matches option A.

ANSWER: A