Question

Consider a model of the hydrogen atom in which an electron orbits a proton in the plane perpendicular to the proton's spin angular momentum (and magnetic dipole moment) at a distance equal to the Bohr radius, \(a_{0}=5.292 \cdot 10^{-11} \mathrm{~m}\). (This is an oversimplified classical model.) The spin of the electron is allowed to be either parallel to the proton's spin or antiparallel to it; the orbit is the same in either case. But since the proton produces a magnetic field at the electron's location, and the electron has its own intrinsic magnetic dipole moment, the energy of the electron differs depending on its spin. The magnetic field produced by the proton's spin may be modeled as a dipole field, like the electric field due to an electric dipole discussed in Chapter 22. Calculate the energy difference between the two electron-spin configurations. Consider only the interaction between the magnetic dipole moment associated with the electron's spin and the field produced by the proton's spin.

Short answer

Answer: To find the energy difference between the parallel and antiparallel electron-spin configurations, follow these steps: 1. Calculate the magnetic field produced by the proton's spin at the electron's location using the dipole magnetic field formula. 2. Find the energies of the electron in both parallel and antiparallel configurations by calculating the magnetic potential energy of the electron. 3. Determine the energy difference by subtracting the antiparallel configuration energy from the parallel configuration energy. By completing these steps and substituting the appropriate expressions, you can calculate the energy difference between the two electron-spin configurations.

Step-by-step solution

Calculate the magnetic field produced by the proton's spin at the electron's location

In order to calculate the magnetic field, we can use the dipole magnetic field formula at a position perpendicular to the dipole moment vector: \(B = \frac{\mu_{0}}{4 \pi} \cdot \frac{2\mu_{p}}{r^{3}}\) Here, \(\mu_{0}\) is the permeability of free space, \(\mu_{p}\) is the magnetic dipole moment of the proton, and \(r\) is the distance between the proton and the electron (which is equal to the Bohr radius, \(a_{0}\)). The magnetic moment of a proton can be given as follows: \(\mu_{p} = g_{p} \frac{e \hbar}{2m_{p}}\) where \(g_{p}\) is the proton's g-factor (\(g_{p} \approx 5.58\)), \(e\) is the elementary charge, \(\hbar\) is the reduced Planck constant, and \(m_{p}\) is the mass of the proton. With these values, we can find the magnetic field at the electron's location.

Find the energy of the electron in its parallel and antiparallel spin configurations

The magnetic potential energy of the electron can be given by the dot product of its magnetic dipole moment, \(\mu_{e}\), and the magnetic field, \(B\): \(U = -\mu_{e} \cdot B\) Since there are two possible spin configurations for the electron (parallel and antiparallel to the proton's magnetic moment), we can find the energies \(E_{\parallel}\) and \(E_{\perp}\) for the two cases. For the parallel configuration, the angle between the magnetic dipole moment and the magnetic field is 0: \(E_{\parallel} = -\mu_{e} \cdot B \cdot \cos{0}\) For the antiparallel configuration, the angle is 180 degrees: \(E_{\perp} = -\mu_{e} \cdot B \cdot \cos{180}\)

Determine the energy difference between the two electron-spin configurations

To find the energy difference, we can subtract the energy of the antiparallel configuration from the parallel configuration: \(\Delta E = E_{\parallel} - E_{\perp}\) By substituting the expressions for \(E_{\parallel}\) and \(E_{\perp}\) and calculating the difference, we can find the energy difference between the two electron-spin configurations.