Rotation operator

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This article concerns the rotation operator, as it appears in quantum mechanics.

The translation operator

The rotation operator R(z, t), with the first argument z indicating the rotation axis and the second t = θ the rotation angle, is based on the translation operator T(a), which is acting on the state |x⟩ in the following manner:

T(a)|x⟩ = |x + a⟩

We have:

T(0) = 1

T(a) T(da)|x⟩ = T(a)|x + da⟩ = |x + a + da⟩ = T(a + da)|x⟩ ⇒

T(a) T(da) = T(a + da)

Taylor development gives:

T(da) = T(0) + dT/da(0) da + ... = 1 - i/h p_x da with p_x = i h dT/da(0)

From that follows:

T(a + da) = T(a) T(da) = T(a)(1 - i/h p_x da) ⇒

[T(a + da) - T(a)]/da = dT/da = - i/h p_x T(a)

This is a differential equation with the solution T(a) = exp(- i/h p_x a).

Additionally, suppose a Hamiltonian H is independent of the x position. Because the translation operator can be written in terms of p_x, and [p_x,H]=0, we know that [H,T(a)]=0. This result means that linear momentum for the system is conserved.

The rotation operator related to the orbital angular momentum

Classically we have l = r x p. This is the same in QM considering r and p as operators. An infinitesimal rotation dt about the z-axis can be expressed by the following infinitesimal translations:

x' = x - y dt

y' = y + x dt

From that follows:

R(z, dt)|r⟩ = R(z, dt)|x, y, z⟩ = |x - y dt, y + x dt, z⟩ = T_x(-y dt) T_y(x dt)|x, y, z⟩ = T_x(-y dt) T_y(x dt)|r⟩

And consequently:

R(z, dt) = T_x(-y dt) T_y(x dt)

Using T_k(a) = exp(- i/h p_k a) with k = x,y and Taylor development we get:

R(z, dt) = exp[- i/h (x p_y - y p_x) dt] = exp(- i/h l_z dt) = 1 - i/h l_z dt + ...

To get a rotation for the angle t, we construct the following differential equation using the condition R(z, 0) = 1:

R(z, t + dt) = R(z, t) R(z, dt) ⇒

[R(z, t + dt) - R(z, t)]/dt = dR/dt = R(z, t) [R(z, dt) - 1]/dt = - i/h l_z R(z, t) ⇒

R(z, t) = exp(- i/h t l_z)

Similar to the translation operator, if we are given a Hamiltonian H which rotationally symmetric about the z axis, [l_z,H]=0 implies [R(z,t),H]=0. This result means that angular momentum is conserved.

For the spin angular momentum about the y-axis we just replace l_z with s_y = h/2 σ_y and we get the spin rotation operator D(y, t) = exp(- i t/2 σ_y).

The effect of the rotation operator on the spin operator and on states

Operators can be represented by matrices. From linear algebra one knows that a certain matrix A can be represented in another base through the basis transformation

A' = P A P^-1

where P is the transformation matrix. If b and c are perpendicular to the y-axis and the angle t lies between them, the spin operator S_b can be transformed into the spin operator S_c through the following transformation:

S_c = D(y, t) S_b D^-1(y, t)

From standard QM we have the known results S_b |b+⟩ = h/2 |b+⟩ and S_c |c+⟩ = h/2 |c+⟩. So we have:

h/2 |c+⟩ = S_c |c+⟩ = D(y, t) S_b D^-1(y, t) |c+⟩ ⇒

S_b D^-1(y, t) |c+⟩ = h/2 D^-1(y, t) |c+⟩

Comparison with S_b |b+⟩ = h/2 |b+⟩ yields |b+⟩ = D^-1(y, t) |c+⟩.

This can be generalized to arbitrary axes.