Orbital Angular Momentum

Figure 9.5: Illustration of a bead of mass $M$ sliding (without friction) on a circular loop of wire of radius $R$ with momentum $\Pi$.

Another type of bound state motion occurs when a particle is constrained to move in a circle. (Imagine a bead sliding on a circular loop of wire, as illustrated in figure 9.5.) We can define $x$ in this case as the path length around the wire and relate it to the angle $\theta$: $x = R \theta$. For a plane wave we have

$$\psi = \exp [i(kx - \omega t)] = \exp [i(kR \theta - \omega t)] .$$ (10.26)

This plane wave differs from the normal plane wave for motion along a Cartesian axis in that we must have $\psi ( \theta ) = \psi ( \theta + 2 \pi )$. This can only happen if the circumference of the loop, $2 \pi R$, is an integral number of wavelengths, i. e., if $2 \pi R / \lambda = m$ where $m$ is an integer. However, since $2 \pi / \lambda = k$, this condition becomes $k R = m$.

Since $\Pi = \hbar k$, the above condition can be written $\Pi_m R = m \hbar$. The quantity

$$L_m \equiv \Pi_m R$$ (10.27)

is called the angular momentum, leading to our final result,

$$L_m = m \hbar , ~~~~ m = 0, \pm 1, \pm 2 , \ldots .$$ (10.28)

We see that the angular momentum can only take on values which are integer multiples of $\hbar$. This represents the quantization of angular momentum, and $m$ in this case is called the angular momentum quantum number. Note that this quantum number differs from the energy quantum number for the particle in the box in that zero and negative values are allowed.

The energy of our bead on a loop of wire can be expressed in terms of the angular momentum:

$$E_m = \frac{\Pi_m^2}{2M} = \frac{L_m^2}{2MR^2} .$$ (10.29)

This means that angular momentum and energy are compatible variables in this case, which further means that angular momentum is a conserved variable. Just as definite values of linear momentum are related to invariance under translations, definite values of angular momentum are related to invariance under rotations. Thus, we have

$$\mbox{invariance under rotation} \Longleftrightarrow \mbox{definite angular momentum}$$ (10.30)

for angular momentum.

We need to briefly address the issue of angular momentum in three dimensions. Angular momentum is actually a vector oriented perpendicular to the wire loop in the example we are discussing. The direction of the vector is defined using a variation on the right-hand rule: Curl your fingers in the direction of motion of the bead around the loop (using your right hand!). The orientation of the angular momentum vector is defined by the direction in which your thumb points. This tells you, for instance, that the angular momentum in figure 9.5 points out of the page.

In quantum mechanics it turns out that it is only possible to measure simultaneously the square of the length of the angular momentum vector and one component of this vector. Two different components of angular momentum cannot be simultaneously measured because of the uncertainty principle. However, the length of the angular momentum vector may be measured simultaneously with one component. Thus, in quantum mechanics, the angular momentum is completely specified if the length and one component of the angular momentum vector are known.

Figure: Illustration of the angular momentum vector $\mbox{\bf L}$ for a tilted loop and its $z$ component $L_z$.

Figure 9.6 illustrates the angular momentum vector associated with a bead moving on a wire loop which is tilted from the horizontal. One component (taken to be the $z$ component) is shown as well. For reasons we cannot explore here, the square of the length of the angular momentum vector $L^2$ is quantized with the following values:

$$L_l^2 = \hbar^2 l (l + 1) , ~~~~~ l = 0,1,2, \ldots .$$ (10.31)

One component (say, the $z$ component) of angular momentum is quantized just like angular momentum in the two-dimensional case, except that $l$ acts as an upper bound on the possible values of $\vert m\vert$. In other words, if the square of the length of the angular momentum vector is $\hbar^2 l (l + 1)$, then the $z$ component can take on the values

$$L_{zm} = \hbar m , ~~~~~ m = -l, -l + 1, \ldots , l - 1, l .$$ (10.32)

The quantity $l$ is called the angular momentum quantum number, while $m$ is called the orientation or magnetic quantum number, the latter for historical reasons.