Stern-Gerlach Experiment - II
The magnetic moment \(\boldsymbol{\mu}\) and the the intrinsic spin angular momentum \(\mathbf{S}\) of a particle are related by the equation,
\begin{equation} \boldsymbol{\mu} = \frac{gq}{2mc}\mathbf{S} \label{eq:isam} \end{equation}
where \(q\) is the unit charge, \(m\) the mass of the particle, and \(c\) the speed of light in vacuum. \(g\) is a dimensionless constant whose value varies with the particle. For example, \(g = 2.00\) for an electron, \(g = 5.58\) for a proton, and \(g = -3.82\) for a neutron1.
It is tempting to presume that the intrinsic spin angular momentum is just the orbital angular momentum of the particle spinning around its own axis. However, spin, as in this context, is a type of angular momentum that has no classical analogue. In fact, Eq. \eqref{eq:isam} can’t be derived from classical arguments, and it is best to think of \(g\) as a dimensionless factor inserted to balance the magnitudes and as well as the units.
In the Stern-Gerlach experiment, we are interested only in the 47th electron in the silver atom. This is because, the spin magnetic moment of the entire atom is effectively due to this single electron. If \(\mathbf{B}\) is the strength of the magnetic field, then the energy of interaction of the magnetic dipole with the external magnetic field is \(- \boldsymbol{\mu} \cdot \mathbf{B}\). The corresponding force experienced by the neutral silver atom is,
\[\begin{aligned} \mathbf{F} & = \mathbf{\nabla (\mu \cdot B)} \\ & = \frac{\partial}{\partial x} \mathbf{\left(\mu \cdot B \right)}\, \hat{\mathbf{i}} + \frac{\partial}{\partial y} \mathbf{\left(\mu \cdot B \right)}\, \hat{\mathbf{j}} + \frac{\partial}{\partial z} \mathbf{\left(\mu \cdot B \right)}\, \hat{\mathbf{k}}. \end{aligned}\]Since the applied magnetic field is in the \(z\) direction, we can consider only the \(z\) component of the force, and neglect the rest. Thus, we have,
\[\begin{equation} \begin{aligned} F_{z} &= \frac{\partial}{\partial z} \mathbf{\left(\mu \cdot B \right)} \\ \nonumber &= \frac{\partial}{\partial z} \left(\mu_{x}B_{x} + \mu_{y}B_{y} + \mu_{z}B_{z} \right) \\ \nonumber & \approx \mu_{z} \frac{\partial B_z}{\partial z}. \end{aligned} \end{equation} \label{eq:force}\]If the magnetic field gradient \(\partial B_z / \partial z\) is negative, the force \(F_{z}\) will be positive for silver atoms with negative \(\mu_{z}\). Such atoms are deflected in the \(+z\) direction.
\(\mu_{z}\), the \(z-\) component of the magnetic moment, can be written as \(\mu_{z} = |\boldsymbol{\mu}| \cos\, \theta\), where \(\theta\) is the angle between the magnetic moment vector (\(\boldsymbol{\mu}\)) and the \(+z\) axis. Intuitively, there are infinitely many values of \(\mu_{z}\) as \(\theta\) can take a continuous range of values. Now, Eq. \eqref{eq:force} can be written as,
\[\begin{equation} F_{z} \approx \mu_{z} \frac{\partial B_z}{\partial z} = \ |\boldsymbol{\mu}| \cos\, \theta\, \frac{\partial B_z}{\partial z}. \end{equation}\]Clearly, \(F_{z}\) has a continuous range of values resulting in a continuous range of deflections. That is, the trail on the screen left by the deflected silver atoms should be a straight line. The extreme points of this line correspond to \(\theta = 0^{\circ}\) and \(\theta = 180^{\circ}\).
What Stern and Gerlach observed was completely different. On the screen were
two clusters of traces, but not a continuous line.
This is a very important result because it tells that the \(z\) component
of the silver atom’s magnetic dipole moment has only two values. Since the
magnetic dipole moment of a neutral silver atom is effectively due to an
electron, this result holds good for electron as well.
Writing only the \(z\) component of Eq \eqref{eq:isam}, we get,
\[\begin{equation} \mu_{z} = \frac{gq}{2mc}S_{z}. \end{equation}\]Calculations show that the two experimentally observed values of \(\mu_{z}\) correspond to \(S_{z} = \pm \hbar/2\).
In summary, Stern-Gerlach experiment showed that the spin angular momentum of an electron has only two values, but not a contious range of values as predicted by the classical theory.
This is why the Stern-Gerlach experiment is a landmark experiment in quantum mechanics.
Reference:
- John S Townsend. A Modern Approach to Quantum Mechanics
-
I am yet to understand how Eq. \eqref{eq:isam} works for a neutral particle like neutron. ↩