Magnetic resonance (quantum mechanics)

Magnetic resonance is a phenomenon that affects a Magnetic dipole when placed in a uniform static magnetic field. Its energy is split into a finite number of energy levels, depending on the value of quantum number of angular momentum. This is similar to energy quantization for atoms, say e− in H atom; in this case the atom, in interaction to an external electric field, transitions between different energy levels by absorbing or emitting photons. Similarly if a magnetic dipole is perturbed with electromagnetic field of proper frequency( $E/{{h}}\,$ ), it can transit between its energy eigenstates, but as the separation between energy eigenvalues is small, the frequency of the photon will be the microwave or radio frequency range. If the dipole is tickled with a field of another frequency, it is unlikely to transition. This phenomenon is similar to that, when a system is acted on by a periodic force of frequency equal to its natural frequency.

Quantum mechanical explanation

As a magnetic dipole, using a spin $\tfrac{1}{2}$ system such as a proton; according to the quantum mechanical state of the system, denoted by : $|\Psi(t)\rangle$ , evolved by the action of an unitary operator $e^{-i{\hat H\hbar}/t}$ ; the result obeys Schrödinger equation:

                         $i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi$

States with definite energy evolve in time with phase $e^{-iEt/\hbar}$ ,( $|\Psi(t)\rangle= |\Psi(0)\rangle e^{-iEt/\hbar}$ ) where E is the energy of the state, since the probability of finding the system in state $| \langle x|\Psi(t)\rangle|^2$ = $| \langle x|\Psi(0)\rangle|^2$ is independent of time. Such states are termed stationary states, so if a system is prepared in a stationary state, (i.e. one of the eigenstates of the Hamiltonian operator), then P(t)=1,i.e. it remains in that state indefinitely. This the case only for isolated systems. When a system in a stationary state is perturbed, its state changes, so it is no longer an eigenstate of the system's complete Hamiltonian. This same phenomenon happens in magnetic resonance for a spin $\tfrac{1}{2}$ system in a magnetic field.

The Hamiltonian for a magnetic dipole $\bold{m}$ (associated with a spin $\tfrac{1}{2}$ particle) in a magnetic field $\bold{B_0}\hat{z}$ is:

                         $\hat{H}=-\bold{m} \bold{B_0} = -\tfrac{\hbar}{2}\gamma \sigma_z \bold{B_0} = -\tfrac{\hbar}{2} \omega_0 \begin{bmatrix}1 & 0 \\0 & -1 \end{bmatrix}$

Here $\omega_0 := \gamma B_0$ is the larmor precession frequency of the dipole for $\bold{B_0}$ magnetic field and $\sigma_z$ is z Pauli matrix. So the eigenvalues of $\hat{H}$ are $-\tfrac{\hbar}{2}\omega_0$ and $\tfrac{\hbar}{2}\omega_0$ . If the system is perturbed by a weak magnetic field $\bold{B_1}$ , rotating counterclockwise in x-y plane (normal to $\bold{B_0}$ ) with angular frequency $\omega$ , so that $\bold{B_1}=\hat{i}B_1 \cos{\omega t}-\hat{j}B_1 \sin{\omega t}$ , then $\begin{bmatrix}1\\0 \end{bmatrix}$ and $\begin{bmatrix}0\\1 \end{bmatrix}$ are not eigenstates of the Hamiltonian, which is modified into

       $\hat{H}=\begin{pmatrix} \bold{B_0} &\bold{B_1}\ e^{ \omega i t}\\ \bold{B_1} e^{- \omega i t} & \bold{-B_0}\end{pmatrix}.$

It is inconvenient to deal with a time-dependent hamiltonian. To make $\hat{H}$ time-independent requires a new reference frame rotating with $\bold{B_1}$ ,i.e. rotation operator $\hat{R}(t)$ on $|\Psi(t)\rangle$ , which amounts to basis change in hilbert space. Using this on Schrödinger's equation, the Hamiltonian becomes:

                $\hat{H^\prime}=R(t)\hat{H} R(t)^\dagger +\tfrac{\hbar}{2}\omega\sigma_z$

Writing $\hat{R(t)}$ in the basis of $\sigma_z$ as-

                $\hat{R}(t)=\begin{pmatrix}e^{-i{\omega}t/2}&0\\0&e^{i{\omega}t/2}\end{pmatrix}$

Using this form of the Hamiltonian a new basis is found:

Plot of Probability Amplitude of Spin Flip at Resonance

Plot of Probability Amplitude of Spin Flip in absence of Resonance

            $\hat{H^\prime}=\tfrac{\hbar}{2}\begin{pmatrix}\Delta\omega&-\omega_1\\-\omega_1&-\Delta\omega\end{pmatrix}$      where  $\Delta\omega=\omega-\omega_ 0$  and  $\omega_1=\gamma B_1$

This Hamiltonian is exactly similar to that of a two state system with unperturbed energies $\tfrac{\hbar}{2}\Delta\omega$ & $-\tfrac{\hbar}{2}\Delta\omega$ with a perturbation expressed by $\tfrac{\hbar}{2}\begin{pmatrix}0&-\omega_1\\-\omega_1&0\end{pmatrix}$ ; According to Rabi oscillation, starting with $\begin{bmatrix}1\\0 \end{bmatrix}$ state, a dipole in parallel to $\bold{B_0}$ with energy $-\tfrac{\hbar}{2}\omega_0$ , the probability that it will transit to $\begin{pmatrix}0\\1\end{pmatrix}.$ state (i.e. it will flip) is

$P_{12}=\frac{|\omega_1^2|}{|\Delta\omega^2+\omega_1^2|}\sin^2[\sqrt{\omega^2+\Delta\omega^2}t/2]$

Now if $\omega = \omega_0$ , i.e. $\bold{B_1}$ rotates with larmor frequency of the dipole in $\bold{B_0}$ magnetic field, then at some point of time, say $t = \frac{(2n+1)\Pi)}{\sqrt{\omega^2+\Delta\omega^2}}$ , it flips into another (previous) energy eigenstate $\begin{bmatrix}0\\1 \end{bmatrix}$ ; i.e. a diplole can be completely flipped. When $\omega \not=\omega_0$ , the probability of change of energy state is small, so the previous case reflected resonance. This resonance condition is used for measurement of the magnetic moment of a dipole, magnetic field at a point in space, etc.

A special case to show applications

A special case occurs where a system oscillates between two unstable levels that have the same life time $\tau$ .^[1] If atoms are excited at a constant, say n/time, to the first state, some decay and the rest have a probability $P_12$ to transition to the second state, so in the time interval between t and (t+dt) the number of atoms that jump to the second state from the first is $n(1-e^{-t/\tau})P_{12}dt$ , so at time t the number of atoms in the second state isL

Plot of Rate of Decay with Variation of Uniform Magnetic Field

B_0

Plot of Half-Width of Lorentz Curve with Changing

B_1

                         $dN=n.e^{-t/\tau}.(1-e^{-t/\tau})P_{12}dt$ 
                                    = $n. e^{-t/\tau} .P_{12} dt$

The rate of decay from state two depends on the number of atoms that were collected in that state from all previous intervals, so the number of atoms in state 2 is $\int_{-\infty}^{0} ne^{-t/\tau}P_{12}\ dt$ ; The rate of decay of atoms from state two is proportional to the number of atoms present in that state, while the constant of proportionality is decay constant $\lambda$ . Performing the integration rate of decay of atoms from state two is obtained as:

                                  $(n/2)\omega^2/(\delta\omega^2+\omega_1^2+1/\tau^2)$

From this expression many interesting points can be exploited, such

Varying uniform magnetic field $B_0$ so that $\omega_0$ in $\delta\omega$ produces a Lorentz curve(see Cauchy distribution), detecting the peak of that curve, the abscissa of it gives $\omega_0$ , so now $\omega$ (angular frequency of rotation of $\bold{B}_1$ = $\gamma$ $(B_0)_{max}$ , so from the known value of $\omega$ and $(B_0)_{max}$ , the gyromagnetic ratio $\gamma$ of the dipole can be measured; by this method we can measure Nuclear spin where all electronic spins are balanced. Correct measurement of nuclear magnetic moment helps to understand the character of nuclear force.
If $\gamma$ is known, by varying $\omega$ , the value of $B_0$ can be obtained. This measurement technique is precise enough for use in sensitive magnetometers. Using this technique, the value of magnetic field acting at a particular lattice site by its environment inside a crystal can be obtained.
By measuring half-width of the curve, d= $\sqrt{\omega_1^2+1/\tau^2}$ , for several values of $\omega_1$ (i.e. of $B_1$ ), we can plot d vs $\omega_1$ , and by extrapolating this line for $\omega_1$ , the lifetime of unstable states can be obtained from the intercept.

Rabi's method

Correspondence between classical and quantum mechanical explanations

Though the notion of spin angular momentum arises only in quantum mechanics and has no classical analogue, magnetic resonance phenomena can be explained via classical physics to some extent. When viewed from the reference frame attached to the rotating field, it seems that the magnetic dipole precesses around a net magnetic field $(\Delta\omega\hat z-\omega_1\hat X)/\gamma$ , where $\hat z$ is the unit vector along uniform magnetic field $B_0$ and $\hat X$ is the same in the direction of rotating field $B_1$ and $\delta\omega=\omega-\omega_0$ .

Proof of Classical Expression for Precession

Pictorial representation of classical Larmor precession

As Classical Electrodynamics tells that torque on a magnetic dipole of moment $\bold{m}$ is $\bold{m}$ × $\bold{B}$ ,so its equation of motion is

$\frac{d\bold{L}}{dt}=$ $\bold{m}$ × $\bold{B}$ , (where $\bold{L}$ is the angular momentum associated with dipole) so -

\frac{d\bold{m}}{dt}= \gamma

$\bold{m}$ ×

\bold{B}

For considered case dipole is under the action of magnetic field $\bold{B}$ and $\bold{B_1}$ , hence

$\frac{d\bold{m}}{dt}= \gamma$ $\bold{m}$ × $\bold{B}+\bold{B_1}$ It is easier to solve it by transforming co-ordinate system to OXYZ in which $\bold{B_1}$ becomes OX axis, in that frame -

$\frac{d\bold{m^'}}{dt}=\frac{d\bold{m}}{dt}+$ $\bold{m}$ × $\bold{\omega}$

here $\bold{\omega}=\omega\hat z$ Using $\omega_0=\gamma B$ and $\omega_1=\gamma B_1$ , one can see that-

\frac{d\bold{m^'}}{dt}=

$\bold{m}$ ×

{(\bold{\omega_0}+\bold{\omega_1}-\bold{\omega})}

=

$\bold{m}$ × (

\bold{\Delta\omega}

+

$\bold{\omega_1}$ )

so, here effective field becomes : $B_{effective} = \Delta\bold{\omega}-\bold{\omega_1} ={\Delta\omega}\bold{\hat z}-\omega_1\bold{\hat X}$

So when $\omega =\omega_0$ , a high precession amplitude allows the magnetic moment to be completely flipped. Classical and quantum mechanical predictions correspond well, which can be viewed as an example of the Bohr Correspondence principle, which states that quantum mechanical phenomena, when predicted in classical regime, should match the classical result. The origin of this correspondence is that the evolution of the expected value of magnetic moment is identical to that obtained by classical reasoning. The expectation value of the magnetic moment is $\langle\bold{m}\rangle= \gamma \langle\bold{S}\rangle$ . The time evolution of $\langle\bold{m}\rangle$ is given by

        $i\hbar\frac{d}{dt }\langle\bold{m}\rangle = \langle[\bold{m},\hat H]\rangle$ 
                $\hat H = -\bold{m}\cdot\bold{B(t)}$

so, $[m_i,\hat H]=[m_i,-m_j B_j]=[\gamma \bold{S}_i,-\gamma \bold{S}_j \bold{B}_j]=-\gamma^2 [\bold{S}_i,\bold{S}_j \bold{B}_j] =-\gamma^2 i\hbar [{\bold{S}_k \bold{B}_j-\bold{S}_k \bold{B}_j}] , (i\neq j,k)$

So, $[m_i,\hat H]=i\hbar \gamma [\bold{B}_j \bold{m}_k -\bold{B}_j \bold{m}_k]$ and

$\frac{d}{dt }\langle\bold{m(t)}\rangle = \gamma\langle\bold{m}(t)\rangle\times\langle\bold{B}(t)\rangle$

which looks exactly similar to the equation of motion of magnetic moment $\bold{m}$ in classical mechanics -

                   $\frac {d}{dt} \bold{m}(t) =\gamma\bold{m}(t)\times\bold{B}(t)$

This analogy in the mathematical equation for the evolution of magnetic moment and its expectation value facilitates to understand the phenomena without a background of quantum mechanics.

Magnetic resonance imaging

Magnetic resonance as a quantum phenomenon

The phenomenon of magnetic resonance is rooted in the existence of spin angular momentum of a quantum system and its specific orientation with respect to an applied magnetic field. Both cases have no explanation in the classical approach and can be understood only by using quantum mechanics. Some people claim that purely quantum phenomena are those that cannot be explained by the classical approach. For example, phenomena in the microscopic domain that can to some extent be described by classical analogy are not really quantum phenomena. Since the basic elements of magnetic resonance have no classical origin, although analogy can be made with Classical Larmor precession, MR should be treated as a quantum phenomenon.

References

↑ Page-449, Chapter-IV, Complement F Quantum Mechanics, Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe

Feynman, Leighton, Sands. The Feynman Lectures on Physics, Volume 3. Narosa Publishing House, New Delhi, 2008.
Cohen-Tannoudji Claude. Quantum Mechanics. Wiley-VCH.
Griffiths David J. An Introduction to Quantum Mechanics. Pearson Education,Inc.

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