Rotation operator (quantum mechanics)

With every physical rotation ${\displaystyle R}$, we postulate a quantum mechanical rotation operator ${\displaystyle D(R)}$ which rotates quantum mechanical states.

$${\displaystyle |\alpha \rangle _{R}=D(R)|\alpha \rangle }$$

In terms of the generators of rotation,

$${\displaystyle D(\mathbf {\hat {n}} ,\phi )=\exp \left(-i\phi {\frac {\mathbf {\hat {n}} \cdot \mathbf {J} }{\hbar }}\right),}$$

where ${\displaystyle \mathbf {\hat {n}} }$ is rotation axis, ${\displaystyle \mathbf {J} }$ is angular momentum, and ${\displaystyle \hbar }$ is the reduced Planck constant.

The rotation operator ${\displaystyle \operatorname {R} (z,\theta )}$, with the first argument ${\displaystyle z}$ indicating the rotation axis and the second ${\displaystyle \theta }$ the rotation angle, can operate through the translation operator ${\displaystyle \operatorname {T} (a)}$ for infinitesimal rotations as explained below. This is why, it is first shown how the translation operator is acting on a particle at position x (the particle is then in the state ${\displaystyle |x\rangle }$ according to Quantum Mechanics).

Translation of the particle at position ${\displaystyle x}$ to position ${\displaystyle x+a}$: ${\displaystyle \operatorname {T} (a)|x\rangle =|x+a\rangle }$

Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator, which does nothing):

$${\displaystyle \operatorname {T} (0)=1}$$

$${\displaystyle \operatorname {T} (a)\operatorname {T} (da)|x\rangle =\operatorname {T} (a)|x+da\rangle =|x+a+da\rangle =\operatorname {T} (a+da)|x\rangle \Rightarrow \operatorname {T} (a)\operatorname {T} (da)=\operatorname {T} (a+da)}$$

Taylor development gives:

$${\displaystyle \operatorname {T} (da)=\operatorname {T} (0)+{\frac {d\operatorname {T} (0)}{da}}da+\cdots =1-{\frac {i}{\hbar }}p_{x}da}$$

with

$${\displaystyle p_{x}=i\hbar {\frac {d\operatorname {T} (0)}{da}}}$$

From that follows:

$${\displaystyle \operatorname {T} (a+da)=\operatorname {T} (a)\operatorname {T} (da)=\operatorname {T} (a)\left(1-{\frac {i}{\hbar }}p_{x}da\right)\Rightarrow {\frac {\operatorname {T} (a+da)-\operatorname {T} (a)}{da}}={\frac {d\operatorname {T} }{da}}=-{\frac {i}{\hbar }}p_{x}\operatorname {T} (a)}$$

This is a differential equation with the solution

$${\displaystyle \operatorname {T} (a)=\exp \left(-{\frac {i}{\hbar }}p_{x}a\right).}$$

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

Classically we have for the angular momentum ${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} .}$ This is the same in quantum mechanics considering ${\displaystyle \mathbf {r} }$ and ${\displaystyle \mathbf {p} }$ as operators. Classically, an infinitesimal rotation ${\displaystyle dt}$ of the vector ${\displaystyle \mathbf {r} =(x,y,z)}$ about the ${\displaystyle z}$-axis to ${\displaystyle \mathbf {r} '=(x',y',z)}$ leaving ${\displaystyle z}$ unchanged can be expressed by the following infinitesimal translations (using Taylor approximation):

$${\displaystyle {\begin{aligned}x'&=r\cos(t+dt)=x-y\,dt+\cdots \\y'&=r\sin(t+dt)=y+x\,dt+\cdots \end{aligned}}}$$

From that follows for states:

$${\displaystyle \operatorname {R} (z,dt)|r\rangle =\operatorname {R} (z,dt)|x,y,z\rangle =|x-y\,dt,y+x\,dt,z\rangle =\operatorname {T} _{x}(-y\,dt)\operatorname {T} _{y}(x\,dt)|x,y,z\rangle =\operatorname {T} _{x}(-y\,dt)\operatorname {T} _{y}(x\,dt)|r\rangle }$$

And consequently:

$${\displaystyle \operatorname {R} (z,dt)=\operatorname {T} _{x}(-y\,dt)\operatorname {T} _{y}(x\,dt)}$$

Using

$${\displaystyle T_{k}(a)=\exp \left(-{\frac {i}{\hbar }}p_{k}a\right)}$$

from above with ${\displaystyle k=x,y}$ and Taylor expansion we get:

$${\displaystyle \operatorname {R} (z,dt)=\exp \left[-{\frac {i}{\hbar }}\left(xp_{y}-yp_{x}\right)dt\right]=\exp \left(-{\frac {i}{\hbar }}L_{z}dt\right)=1-{\frac {i}{\hbar }}L_{z}dt+\cdots }$$

with ${\displaystyle L_{z}=xp_{y}-yp_{x}}$ the ${\displaystyle z}$-component of the angular momentum according to the classical cross product.

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

$${\displaystyle {\begin{aligned}&\operatorname {R} (z,t+dt)=\operatorname {R} (z,t)\operatorname {R} (z,dt)\\[1.1ex]\Rightarrow {}&{\frac {d\operatorname {R} }{dt}}={\frac {\operatorname {R} (z,t+dt)-\operatorname {R} (z,t)}{dt}}=\operatorname {R} (z,t){\frac {\operatorname {R} (z,dt)-1}{dt}}=-{\frac {i}{\hbar }}L_{z}\operatorname {R} (z,t)\\[1.1ex]\Rightarrow {}&\operatorname {R} (z,t)=\exp \left(-{\frac {i}{\hbar }}\,t\,L_{z}\right)\end{aligned}}}$$

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

For the spin angular momentum about for example the ${\displaystyle y}$-axis we just replace ${\displaystyle L_{z}}$ with ${\textstyle S_{y}={\frac {\hbar }{2}}\sigma _{y}}$ (where ${\displaystyle \sigma _{y}}$ is the Pauli Y matrix) and we get the spin rotation operator

$${\displaystyle \operatorname {D} (y,t)=\exp \left(-i{\frac {t}{2}}\sigma _{y}\right).}$$

Operators can be represented by matrices. From linear algebra one knows that a certain matrix ${\displaystyle A}$ can be represented in another basis through the transformation

$${\displaystyle A'=PAP^{-1}}$$

where ${\displaystyle P}$ is the basis transformation matrix. If the vectors ${\displaystyle b}$ respectively ${\displaystyle c}$ are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle ${\displaystyle t}$ between them. The spin operator ${\displaystyle S_{b}}$ in the first basis can then be transformed into the spin operator ${\displaystyle S_{c}}$ of the other basis through the following transformation:

$${\displaystyle S_{c}=\operatorname {D} (y,t)S_{b}\operatorname {D} ^{-1}(y,t)}$$

From standard quantum mechanics we have the known results ${\textstyle S_{b}|b+\rangle ={\frac {\hbar }{2}}|b+\rangle }$ and ${\textstyle S_{c}|c+\rangle ={\frac {\hbar }{2}}|c+\rangle }$ where ${\displaystyle |b+\rangle }$ and ${\displaystyle |c+\rangle }$ are the top spins in their corresponding bases. So we have:

$${\displaystyle {\frac {\hbar }{2}}|c+\rangle =S_{c}|c+\rangle =\operatorname {D} (y,t)S_{b}\operatorname {D} ^{-1}(y,t)|c+\rangle \Rightarrow }$$

$${\displaystyle S_{b}\operatorname {D} ^{-1}(y,t)|c+\rangle ={\frac {\hbar }{2}}\operatorname {D} ^{-1}(y,t)|c+\rangle }$$

Comparison with ${\textstyle S_{b}|b+\rangle ={\frac {\hbar }{2}}|b+\rangle }$ yields ${\displaystyle |b+\rangle =D^{-1}(y,t)|c+\rangle }$.

This means that if the state ${\displaystyle |c+\rangle }$ is rotated about the ${\displaystyle y}$-axis by an angle ${\displaystyle t}$, it becomes the state ${\displaystyle |b+\rangle }$, a result that can be generalized to arbitrary axes.

• Symmetry in quantum mechanics
• Spherical basis
• Optical phase space

• L.D. Landau and E.M. Lifshitz: Quantum Mechanics: Non-Relativistic Theory, Pergamon Press, 1985
• P.A.M. Dirac: The Principles of Quantum Mechanics, Oxford University Press, 1958
• R.P. Feynman, R.B. Leighton and M. Sands: The Feynman Lectures on Physics, Addison-Wesley, 1965