Date: Fri, 21 June 2024 06:22:55 +00:00
Mime-Version: 1.0 (Produced by Tiki)
Content-Type: application/x-tikiwiki;
pagename=Precession-quantum%20mechanical%20description;
flags="";
author=kettinger.adam;
version=3;
lastmodified=1410169083;
author_id=10.11.12.184;
summary="";
hits=2484;
description="";
charset=utf-8
Content-Transfer-Encoding: binary
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, {EQUATION()}$J^2${EQUATION} the possible eigenvalues are
::{EQUATION()}
\label{I2_eigenvalues}
J^2 = \hbar^2 I(I+1)
{EQUATION}(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 {EQUATION()}$\mathbf{e_z}${EQUATION}:
::{EQUATION()}
\label{I_z_eigenvalues}
J_z = \hbar m, \hspace{20pt} -I \leq m \leq I
{EQUATION}(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
::{EQUATION()}
\label{gyro_ratio_quantum}
\boldsymbol{\mu} = \gamma \mathbf{J}
{EQUATION}(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'):
::{EQUATION()}
\label{matrix_element}
\big < I , m \big | \mu_x \big | I , m' \big >= \gamma \big < I , m \big | J_x \big | I , m' \big >
{EQUATION}(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:
::{EQUATION()}
\label{Zeeman_Hamiltonian}
H_{Zeeman} = - \gamma \hbar \mathbf{JB}
{EQUATION}(5)::
If the external magnetic fields is $B_0$ then the splitted energy levels are
::{EQUATION()}
\label{Zeeman_energies}
E_m = - \gamma \hbar B_0 m = - \hbar m \omega_0
{EQUATION}(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
::{EQUATION()}
\label{general_wavefunction}
\Psi (t) = \sum_{m= -I}^{I} { C_m \big | I , m \big > \mathrm{e}^{ - \tfrac{ \mathrm{i}}{\hbar} E_m t} }
{EQUATION}(7)::
Using equations (4) and (7) we can compute the mean value of {EQUATION()}$\mu_x${EQUATION} for a spin-half particle, exploiting the fact that in this case {EQUATION()}$I = \frac{1}{2}${EQUATION}.
::{EQUATION()}
\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 }
{EQUATION}(8)::
Now we express the {EQUATION()}$x${EQUATION} component of the angular momentum with the usual ladder operators
::{EQUATION()}
\label{ladder_operators1}
J^+ = J_x + \mathrm{i}J_y
{EQUATION}(9)::
::{EQUATION()}
J^- = J_x - \mathrm{i}J_y
{EQUATION}(10)::
::{EQUATION()}
\label{ladder_operators2}
J^+ \big | I , m \big > = \sqrt{ I ( I+1 ) - m ( m+1 ) } \big | I , m+1 \big >
{EQUATION}(11)::
::{EQUATION()}
J^- \big | I , m \big > = \sqrt{ I ( I+1 ) - m ( m-1 ) } \big | I , m-1 \big >
{EQUATION}(12)::
::{EQUATION()}
J_x = \frac{1}{2} \Big (J^+ + J^- \Big )
\label{ladder_operators3}
{EQUATION}(13)::
::{EQUATION()}
J_y = \frac{1}{2 \mathrm{i}} \Big (J^+ - J^- \Big )
{EQUATION}(14)::
After that substitute (6), and (11)-(14) in (8), consider the fact that both {EQUATION()}$m${EQUATION} and {EQUATION()}$m'${EQUATION} can only take the values of {EQUATION()}$\frac{1}{2}${EQUATION} and {EQUATION()}$ - \frac{1}{2}${EQUATION}, and that eigenstates with different {EQUATION()}$m${EQUATION} values are orthogonal to each other:
::{EQUATION()}
\label{mu_x_mean_value2}
\big < \mu_x \big > = \frac{1}{2} V \gamma \hbar \bigg ( C^*_{ \frac{1}{2} } C_{ - \frac{1}{2} } \mathrm{e}^{- \mathrm{i} \omega_0 t } + C_{ \frac{1}{2} } C^*_{ - \frac{1}{2} } \mathrm{e}^{ \mathrm{i} \omega_0 t } \bigg ) = V \gamma \hbar \hspace{2pt} \Re \bigg ( C^*_{ \frac{1}{2} } C_{ - \frac{1}{2} } \mathrm{e}^{ - \mathrm{i} \omega_0 t } \bigg )
{EQUATION}(15)::
Without loss of generality we can assume that {EQUATION()}$C_{ \frac{1}{2} } = a \mathrm{e}^{ \mathrm{i} \alpha}${EQUATION} and {EQUATION()}$C_{ - \frac{1}{2} } = b \mathrm{e}^{ \mathrm{i} \beta}${EQUATION} for some {EQUATION()}$ a, b, \alpha, \beta${EQUATION} real numbers. Furthermore, the normalization criterion requires that {EQUATION()}$ | C_{ \frac{1}{2} } |^2 + | C_{ -\frac{1}{2} } |^2 = \frac{1}{V} ${EQUATION}, which allowes us to write {EQUATION()}$a${EQUATION} and {EQUATION()}$b${EQUATION} as {EQUATION()}$ a = \frac{1}{\sqrt{V}} \mathrm{cos} \big ( \frac{\theta}{2} \big )${EQUATION} and {EQUATION()}$ b = \frac{1}{\sqrt{V}} \mathrm{sin} \big ( \frac{\theta}{2} \big ) ${EQUATION}. Using these expressions (15) becomes the following:
::{EQUATION()}
\label{mu_x_mean_value2}
\big < \mu_x \big > = V \gamma \hbar a b \hspace{2pt} \mathrm{cos} \big ( \alpha - \beta + \omega_0 t) = \frac{\gamma \hbar}{2} \mathrm{sin} (\theta) \mathrm{cos} \big (\alpha - \beta - \omega_0 t \big )
{EQUATION}(16)::
In an absolutely similar manner one can show that the mean values of the other two components of magnetic moment will be the following:
::{EQUATION()}
\label{mu_x_mean_value3}
\big < \mu_y \big > = \frac{\gamma \hbar}{2} \mathrm{sin} (\theta) \mathrm{sin} \big (\alpha - \beta - \omega_0 t \big )
{EQUATION}(17)::
::{EQUATION()}
\label{mu_x_mean_value4}
\big < \mu_z \big > = \frac{\gamma \hbar}{2} \mathrm{cos} (\theta)
{EQUATION}(18)::
With these expressions we have presented that the quantum mechanical calculations of a spin-half particle also shows the precession movement in the sense of mean values as equations (16), (17) and (18) describe a vector that precesses around axis {EQUATION()}$z${EQUATION} with angular frequency {EQUATION()}$\omega_0 = - \gamma B_0${EQUATION}.