Q5Q - Fundamentals Of Physics

Question

An electron in an external magnetic field ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$. has its spin angular momentum S_z antiparallel to ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$. If the electron undergoes a spin-flip so that S_z is then parallel th ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$, must energy be supplied to or lost by the electron?

Step-by-Step Solution

Answer

Energy must be supplied to the electron if the electron undergoes a spin-flop.

Step 1: Given

An electron in an external magnetic field ${\stackrel{\rightharpoonup }{B}}_{ext}$ has its spin angular momentum S_z  antiparallel to ${\stackrel{\rightharpoonup }{B}}_{ext}$.

Step 2: Determining the concept

Using the formula for the potential for the electron and inserting the spin angular magnetic momentum, it can be predicted whether energy should be supplied to or lost by the electron if the electron undergoes a spin-flop.

The formulae are as follows:

$\begin{array}{rcl}U& =& -\stackrel{\rightharpoonup }{{\mu }_s}.{\stackrel{\rightharpoonup }{B}}_{ext} .\\ {\mu }_{s,z}& =& -\frac{e}{m}{S}_z .\\ S_z& =& m_s\frac{h}{2\mathrm{\pi }} .\end{array}$

Step 3: Determining whether energy should be supplied to or lost by the electron if the electron undergoes a spin-flop

The potential energy for the electron is given by,

$$\begin{gathered} U=-{\mu }_{s,z}.B_{ext ,} \\ And , \\ {\mu }_{s,z}=-\frac{e}{m}{S}_z , \end{gathered}$$

But,

$S_z=m_s\frac{h}{2\mathrm{\pi }} ,$

When the spin angular momentum S_z is parallel to ${\stackrel{\rightharpoonup }{B}}_{ext}$, then $m_s=+\frac{1}{2}$, and when it is antiparallel ${\stackrel{\rightharpoonup }{B}}_{ext}$ to, $m_s=-\frac{1}{2}$.

Hence, the potential energy of the electron when its spin angular momentum S_z is parallel to ${\stackrel{\rightharpoonup }{B}}_{ext}$ after spin-flop is,

$U=\frac{e}{m}{S}_z.B_{ext} ,$

This implies that energy must be supplied to the electron if the electron undergoes a spin-flop.

Whether energy is supplied to or lost by an electron if the electron undergoes a spin-flop depends on the sign of the spin magnetic quantum number.