91影视

Magnetic field,$B_{ext}=0.500\mathrm{T}$

To calculate the $m_l$values, we can use the formula of energy associated with the orientation of an orbital magnetic dipole moment in an external field. Substituting the values of $鈭�m_l$, Bohr magnetron, and the magnetic field, we can calculate the spacing between the energy levels.

Formula:

$U=-m_l{渭}_B{B}_{ext}$

The energyU associated with the orientation of the orbital magnetic dipole moment in an external magnetic field${\stackrel{鈫�}{B}}_{ext}$is

$\begin{array}{rcl}U& =& -{\stackrel{鈫�}{渭}}_{orb}.{\stackrel{鈫�}{B}}_{ext}\\ & =& -{渭}_{orb,z}{B}_{ext}\end{array}$

Since

${渭}_{orb,z}=-m_l\frac{eh}{4蟺m}$

Also,

$\frac{eh}{4蟺m}={渭}_B$

Thus,

${渭}_{orb,z}=-m_l{渭}_B$

Therefore,

$\begin{array}{rcl}U& =& -{\stackrel{鈫�}{渭}}_{orb}.{\stackrel{鈫�}{B}}_{ext}\\ & =& -m_l{渭}_B{B}_{ext}\end{array}$

From the figure, we can observe that there is no splitting of energy level$E_1$ in the external magnetic field.

Hence its energy does not change by the application of external magnetic field${\stackrel{鈫�}{B}}_{ext}$

Hence$U鈭�m_l=0$

$鈭�m_l=0$

The energy level$E_2$ splits into three energy levels, with the middle one unshifted from the original value of energy. $E_2$Hence its$m_l$value is zero.

In other two lines, one line is shifted above the unshifted line and another one is shifted below.

Hence the upper line has$m_l=+1$and lower will be$m_l=-1$

The spacing between the energy levels is given by

$\begin{array}{rcl}螖U& =& {渭}_B{B}_{ext}\\ & =& \left|螖\left(-m_l{渭}_B{B}_{ext}\right)\right|\end{array}$

For any pair of adjacent levels in triplet,

$\left|螖m_l\right|=1$

Therefore,

where,${渭}_B$ is Bohr magnetron, and its value is${渭}_B=9.27脳{10}^{-24}\mathrm{J}/\mathrm{T}$

$\begin{array}{rcl}螖U& =& 9.27脳{10}^{-24}\frac{\mathrm{J}}{\mathrm{T}}脳0.500\mathrm{T}\\ & =& 4.64脳{10}^{-24}\mathrm{J}\end{array}$

Amount of energy in joules represented by the spacing between the triplet levels is $4.64脳{10}^{-24}\mathrm{J}$.