Precession-quantum mechanical description

In quantum mechanics both angular and magnetic moment are represented by hermitic operators characterized by their eigenvalues and eigenstates. For the square of the angular momentum, $J^2$ the possible eigenvalues are

$\label{I2_eigenvalues} J^2 = \hbar^2 I(I+1)$ (1)

Where I is the integer or half-integer quantum number of the angular momentum. For the projection of the angular momentum to a specific directon, for example to $\mathbf{e_z}$ :

$\label{I_z_eigenvalues} J_z = \hbar m, \hspace{20pt} -I \leq m \leq I$ (2)

For a single spin-half particle, there is a similar relation between angular and magnetic moment as it was in the classical case, that is

$\label{gyro_ratio_quantum} \boldsymbol{\mu} = \gamma \mathbf{J}$ (3)

Therefore the matrix elements of a specific component of the magnetic moment can be computed using the angular momentum operator. With the bra-ket notation introduced by Dirac, the matrix element with the states described by quantum numbers (I, m) and (I, m'):

$\label{matrix_element} \big < I , m \big | \mu_x \big | I , m' \big >= \gamma \big < I , m \big | J_x \big | I , m' \big >$ (4)

If the particle is placed into external magnetic field, its originally degenerated spectrum splits based on the projection of the magnetic moment to the external field - this is called the Zeeman effect. The descriptive Hamiltonian:

$\label{Zeeman_Hamiltonian} H_{Zeeman} = - \gamma \hbar \mathbf{JB}$ (5)

If the external magnetic fields is $B_0$ then the splitted energy levels are

$\label{Zeeman_energies} E_m = - \gamma \hbar B_0 m = - \hbar m \omega_0$ (6)

In general, the wavefunction of the particle is a sum of the eigenstates with different weights, each multiplied by the time propagation term with one of the energies described above

$\label{general_wavefunction} \Psi (t) = \sum_{m= -I}^{I} { C_m \big | I , m \big > \mathrm{e}^{ - \tfrac{ \mathrm{i}}{\hbar} E_m t} }$ (7)

Using equations (4) and (7) we can compute the mean value of $\mu_x$ for a spin-half particle, exploiting the fact that in this case $I = \frac{1}{2}$ .

$\label{mu_x_mean_value1} \big < \mu_x \big > = \big < \Psi (t) \big | \mu_x \big | \Psi (t) \big > = V \sum_{m,m'} \gamma \hbar C^*_{m'} C_m \big < m' \big | J_x \big | m \big > \mathrm{e}^{ \tfrac {\mathrm{i}} {\hbar} \big ( E_{m'} - E_m \big ) t }$ (8)

Now we express the $x$ component of the angular momentum with the usual ladder operators

$\label{ladder_operators1} J^+ = J_x + \mathrm{i}J_y$ (9)

$J^- = J_x - \mathrm{i}J_y$ (10)

$\label{ladder_operators2} J^+ \big | I , m \big > = \sqrt{ I ( I+1 ) - m ( m+1 ) } \big | I , m+1 \big >$ (11)

$J^- \big | I , m \big > = \sqrt{ I ( I+1 ) - m ( m-1 ) } \big | I , m-1 \big >$ (12)

$J_x = \frac{1}{2} \Big (J^+ + J^- \Big ) \label{ladder_operators3}$ (13)

$J_y = \frac{1}{2 \mathrm{i}} \Big (J^+ - J^- \Big )$ (14)

After that substitute (6), and (11)-(14) in (8), consider the fact that both $m$ and $m'$ can only take the values of $\frac{1}{2}$ and $- \frac{1}{2}$ , and that eigenstates with different $m$ values are orthogonal to each other:

$\label{mu_x_mean_value2} \big < \mu_x \big > = \frac{1}{2} V \gamma \hbar \bigg ( C^*_{ \frac{1}{2} } C_{ - \frac{1}{2} } \mathrm{e}^{- \mathrm{i} \omega_0 t } + C_{ \frac{1}{2} } C^*_{ - \frac{1}{2} } \mathrm{e}^{ \mathrm{i} \omega_0 t } \bigg ) = V \gamma \hbar \hspace{2pt} \Re \bigg ( C^*_{ \frac{1}{2} } C_{ - \frac{1}{2} } \mathrm{e}^{ - \mathrm{i} \omega_0 t } \bigg )$ (15)

Without loss of generality we can assume that $C_{ \frac{1}{2} } = a \mathrm{e}^{ \mathrm{i} \alpha}$ and $C_{ - \frac{1}{2} } = b \mathrm{e}^{ \mathrm{i} \beta}$ for some $a, b, \alpha, \beta$ real numbers. Furthermore, the normalization criterion requires that $| C_{ \frac{1}{2} } |^2 + | C_{ -\frac{1}{2} } |^2 = \frac{1}{V}$ , which allowes us to write $a$ and $b$ as $a = \frac{1}{\sqrt{V}} \mathrm{cos} \big ( \frac{\theta}{2} \big )$ and $b = \frac{1}{\sqrt{V}} \mathrm{sin} \big ( \frac{\theta}{2} \big )$ . Using these expressions (15) becomes the following:

$\label{mu_x_mean_value2} \big < \mu_x \big > = V \gamma \hbar a b \hspace{2pt} \mathrm{cos} \big ( \alpha - \beta + \omega_0 t) = \frac{\gamma \hbar}{2} \mathrm{sin} (\theta) \mathrm{cos} \big (\alpha - \beta - \omega_0 t \big )$ (16)

In an absolutely similar manner one can show that the mean values of the other two components of magnetic moment will be the following:

$\label{mu_x_mean_value3} \big < \mu_y \big > = \frac{\gamma \hbar}{2} \mathrm{sin} (\theta) \mathrm{sin} \big (\alpha - \beta - \omega_0 t \big )$ (17)

$\label{mu_x_mean_value4} \big < \mu_z \big > = \frac{\gamma \hbar}{2} \mathrm{cos} (\theta)$ (18)

With these expressions we have presented that the quantum mechanical calculations of a spin-half particle also shows the precession movement in the sense of mean values as equations (16), (17) and (18) describe a vector that precesses around axis $z$ with angular frequency $\omega_0 = - \gamma B_0$ .

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Precession-quantum mechanical description

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Medical Imaging

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