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 electronspin 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: The energy difference between the two electron spin configurations is approximately 4.62 x 10^-33 J.

Step-by-step solution

Find the magnetic field produced by the proton

To find the magnetic field produced by the proton at the electron's location, we can use the formula for the magnetic field due to a magnetic dipole: \(B = \frac{\mu_0}{4 \pi} \frac{1}{r^3} (2 \mu_p \cos \theta + \mu_p \sin \theta)\) where \(\mu_0\) is the vacuum permeability, \(r\) is the distance between the dipole and the point where the field is being calculated, \(\mu_p\) is the magnetic dipole moment of the proton, and \(\theta\) is the angle between the dipole moment and the position vector. In our case, the electron is orbiting in the plane perpendicular to the proton's spin angular momentum, so \(\theta = 90^\circ\). Therefore, \(\cos \theta = 0\) and \(\sin \theta = 1\). So the magnetic field is given by: \(B = \frac{\mu_0}{4 \pi} \frac{\mu_p}{r^3}\)

Plug in the given values

We are given that the Bohr radius is \(a_0 = 5.292 \times 10^{-11} \mathrm{~m}\), which is our distance \(r\). The magnetic dipole moment of the proton, \(\mu_p\), can be found using the nuclear magneton (\(\mu_N\)), which is equal to \(5.05 \times 10^{-27} \mathrm{~J/T}\). For the proton, \(\mu_p\) is equal to the nuclear magneton, so \(\mu_p = 5.05 \times 10^{-27} \mathrm{~J/T}\). Now we can plug these values into the formula for the magnetic field: \(B = \frac{\mu_0}{4 \pi} \frac{5.05 \times 10^{-27}}{(5.292 \times 10^{-11})^3} = 2.49 \times 10^{-10} \mathrm{~T}\)

Calculate the energy difference

Now that we have the magnitude of the magnetic field produced by the proton, we can find the energy difference between the two electron spin configurations. The energy difference is given by the dot product of the electron's magnetic dipole moment and the magnetic field: \(\Delta E = -\mu_e \cdot B\) The electron's magnetic dipole moment, \(\mu_e\), can be found using the Bohr magneton (\(\mu_B\)), which is equal to \(9.27 \times 10^{-24} \mathrm{~J/T}\). The magnetic dipole moment associated with the electron's spin is either parallel or antiparallel to the magnetic field, so the dot product is either equal to the product of their magnitudes or minus the product of their magnitudes: \(\Delta E = -(\mu_e B - (-\mu_e B)) = 2 \mu_e B\) Plugging in the values we have found, we get: \(\Delta E = 2 (9.27 \times 10^{-24}) (2.49 \times 10^{-10}) = 4.62 \times 10^{-33} \mathrm{~J}\) So the energy difference between the two electron spin configurations is approximately \(4.62 \times 10^{-33} \mathrm{~J}\).

Key concepts

Bohr Radius

The Bohr radius, denoted as \(a_0\), is a fundamental physical constant that represents the mean distance between the proton and the electron in the ground state of a hydrogen atom. Its value is approximately \(5.292 \times 10^{-11} \) meters. This value is derived from the principles of Niels Bohr's atomic model, which, despite its simplicity compared to modern quantum mechanics, provides insightful explanations for the quantized energy levels of electrons in an atom.

In the context of the exercise, the given value of the Bohr radius was used to calculate the magnetic field at the electron's location due to the proton's magnetic dipole moment. This is because the electron, as per the exercise scenario, orbits the proton at a distance equal to the Bohr radius. Recognizing the role of the Bohr radius is critical in understanding atomic structure and the forces that govern the electron's movement around the nucleus.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that measures the strength and orientation of a magnetic source. For charged particles such as electrons and protons, the magnetic dipole moment arises due to their intrinsic spin and movement through space. The magnitude of the proton's magnetic dipole moment is represented by \(\mu_p\), and it plays a crucial role in generating a magnetic field that interacts with other magnetic dipoles in its vicinity.

During the solved exercise, the magnetic dipole moment of the proton is used to determine the strength of the magnetic field experienced by the electron in its orbit. The influence of this magnetic field alters the energy levels of the electron depending on its own magnetic dipole moment and spin configuration relative to the proton's field.

Spin Angular Momentum

Spin angular momentum refers to the intrinsic form of angular momentum carried by elementary particles, such as electrons and protons, due to their spin. Unlike the angular momentum associated with objects in macroscopic rotational motion, spin angular momentum is a quantum property and does not correspond to actual spinning in physical space.

For the exercise at hand, the concept of spin angular momentum is crucial because the magnetic dipole moment of the proton is aligned with its spin angular momentum. Moreover, the energy difference calculated for the electron in the hydrogen atom model depends on whether the electron's spin is parallel or anti-parallel to the proton's spin angular momentum. This affects the orientation and strength of the electron's magnetic dipole moment in the magnetic field created by the proton.

Magnetic Field of Proton

The magnetic field of a proton is generated by its magnetic dipole moment and can be mathematically described using the principles of electromagnetism. This field resembles that of a bar magnet, with a direction that points from the proton's magnetic south pole to its magnetic north pole. The field's strength and orientation vary depending on the distance from the proton and the angle in relation to the proton's spin.

In the context of our problem, the magnetic field produced by the proton's magnetic dipole moment at the location of the orbiting electron directly affects the electron's potential energy in its spin states. Calculating the strength of this field at the Bohr radius allows students to understand the interaction between the electron and the proton in a hydrogen atom.

Electron Spin Configurations

Electron spin configurations relate to the possible orientations of an electron's spin angular momentum with respect to an external magnetic field. In basic terms, the spin can be either parallel (aligned in the same direction) or antiparallel (aligned in the opposite direction) to the field. These configurations are integral to determining the magnetic properties of atoms and the energy differences observed in magnetic interactions.

The exercise provided illustrates how these spin configurations influence the electron's energy state within a magnetic field generated by the proton's spin. The energy difference between parallel and antiparallel spins, derived using the electron's magnetic dipole moment and the calculated magnetic field strength, is essential for understanding phenomena like the Zeeman effect, magnetic resonance imaging (MRI), and other applications in quantum physics and chemistry.