Magnetic resonance (quantum mechanics)

Magnetic resonance is a quantum mechanical resonant effect that can appear when a magnetic dipole is exposed to a static magnetic field and perturbed with another, oscillating electromagnetic field. Due to the static field, the dipole can assume a number of discrete energy eigenstates, depending on the value of its angular momentum quantum number. The oscillating field can then make the dipole transit between its energy states with a certain probability and at a certain rate. The overall transition probability will depend on the field's frequency and the rate will depend on its amplitude. When the frequency of that field leads to the maximum possible transition probability between two states, a magnetic resonance has been achieved. In that case, the energy of the photons composing the oscillating field matches the energy difference between said states. If the dipole is tickled with a field oscillating far from resonance, it is unlikely to transition. That is analogous to other resonant effects, such as with the forced harmonic oscillator. The periodic transition between the different states is called Rabi cycle and the rate at which that happens is called Rabi frequency. The Rabi frequency should not be confused with the field's own frequency. Since many atomic nuclei species can behave as a magnetic dipole, this resonance technique is the basis of nuclear magnetic resonance, including nuclear magnetic resonance imaging and nuclear magnetic resonance spectroscopy.
Quantum mechanical explanation.
As a magnetic dipole, using a spin formula_1 system such as a proton; according to the quantum mechanical state of the system, denoted by :formula_2 , evolved by the action of a unitary operator formula_3; the result obeys Schrödinger equation:
 formula_4
States with definite energy evolve in time with phase formula_5 ,( formula_6 ) where E is the energy of the state, since the probability of finding the system in state formula_7= formula_8 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 formula_1 system in a magnetic field.
The Hamiltonian for a magnetic dipole formula_10 (associated with a spin formula_1 particle) in a magnetic field formula_12 is: 
 formula_13
Here formula_14 is the larmor precession frequency of the dipole for formula_15 magnetic field and formula_16 is z Pauli matrix. So the eigenvalues of formula_17 are formula_18 and formula_19. If the system is perturbed by a weak magnetic field formula_20, rotating counterclockwise in x-y plane (normal to formula_15) with angular frequency formula_22, so that formula_23 , then formula_24 and formula_25 are not eigenstates of the Hamiltonian, which is modified into 
 formula_26
It is inconvenient to deal with a time-dependent hamiltonian. To make formula_17 time-independent requires a new reference frame rotating with formula_20, i.e. rotation operator formula_29 on formula_2, which amounts to basis change in Hilbert space. Using this on Schrödinger's equation, the Hamiltonian becomes:
 formula_31
Writing formula_32 in the basis of formula_16 as-
 formula_34
Using this form of the Hamiltonian a new basis is found: 
 formula_35 where formula_36 and formula_37 
This Hamiltonian is exactly similar to that of a two state system with unperturbed energies formula_38 & formula_39 with a perturbation expressed by formula_40; According to Rabi oscillation, starting with formula_24 state, a dipole in parallel to formula_15 with energy formula_18, the probability that it will transit to formula_44 state (i.e. it will flip) is
Now consider formula_45, i.e. the formula_20 field oscillates at the same rate the dipole exposed to the formula_15 field does. That is a case of resonance. Then at specific points in time, namely formula_48, the dipole will flip, going to the other energy eigenstate formula_25 with a 100% probability. When formula_50, the probability of change of energy state is small. Therefore, the resonance condition can be used, for instance, to measure the magnetic moment of a dipole or the magnetic field at a point in space.
A special case to show applications.
A special case occurs where a system oscillates between two unstable levels that have the same life time formula_51 . If atoms are excited at a constant, say n/time, to the first state, some decay and the rest have a probability formula_52 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 formula_53, so at time t the number of atoms in the second state is
 formula_54
 =formula_55 
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 formula_56; 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 formula_57 . Performing the integration rate of decay of atoms from state two is obtained as:
 formula_58
From this expression many interesting points can be exploited, such 
Rabi's method.
The existence of spin angular momentum of electrons was discovered experimentally by the Stern–Gerlach experiment. In that study a beam of neutral atoms with one electron in the valence shell, carrying no orbital momentum (from the viewpoint of quantum mechanics) was passed through an inhomogeneous magnetic field. This process was not approximate due to the small deflection angle, resulting in considerable uncertainty in the measured value of the split beam.
Rabi's method was an improvement over Stern-Gerlach. As shown in the figure, the source emits a beam of neutral atoms, having spin angular momentum formula_78. The beam passes through a series of three aligned magnets. Magnet 1 produces an inhomogeneous magnetic field with a high gradientformula_79 (as in Stern-Gerlach), so the atoms having 'upward' spin (with formula_80) will deviate downward (path 1), i.e. to the region of less magnetic field B, to minimize energy. Atoms with 'downward' spin with formula_81) will deviate upward similarly (path 2). Beams are passed through slit 1, to reduce any effects of source beyond. Magnet 2 produces only a uniform magnetic field in the vertical direction applying no force on the atomic beam, and magnet 3 is actually inverted magnet 1. In the region between the poles of magnet 3, atoms having 'upward' spin get upward push and atoms having 'downward' spin feel downward push, so their path remains 1 and 2 respectively. These beams pass through a second slit S2, and arrive at detector and get detected. 
If a horizontal rotating field formula_75, angular frequency of rotation formula_74 is applied in the region between poles of magnet 2, produced by oscillating current in circular coils then there is a probability for the atoms passing through there from one spin state to another (formula_84 and vice versa), when formula_74=formula_86, Larmor frequency of precession of magnetic moment in B. The atoms that transition from 'upward' to 'downward' spin will experience a downward force while passing through magnet 3, and will follow path 1'. Similarly, atoms that change from 'downward' to 'upward' spin will follow path 2', and these atoms will not reach the detector, causing a minimum in detector count. If angular frequency formula_74 of formula_75 is varied continuously, then a minimum in detector current will be obtained (when formula_74=formula_86). From this known value of formula_74(formula_92, where g is 'Landé g factor'), 'Landé g factor' is obtained which will enable one to have correct value of magnetic moment formula_93. This experiment, performed by Isidor Isaac Rabi is more sensitive and accurate compared than Stern-Gerlach.
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 formula_94, where formula_95 is the unit vector along uniform magnetic field formula_59 and formula_97is the same in the direction of rotating field formula_75 and formula_99.
 
So when formula_100, 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 formula_101 . The time evolution of formula_102 is given by 
 formula_103
 formula_104
so, formula_105
So, 
formula_106 and
which looks exactly similar to the equation of motion of magnetic moment formula_10 in classical mechanics -
 formula_108 
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.
In magnetic resonance imaging (MRI) the proton's spin angular momentum is used. The most available source for protons is the hydrogen atoms in water. A strong magnetic field formula_109 applied to water causes the appearance of two different energy levels for spin angular momentum, -(formula_110) and -(formula_111), using formula_112 .
According to the Boltzmann distribution (the number of systems having energy formula_113 out of formula_114 at temperature formula_115) is formula_116 (where k is Boltzmann constant) lower energy level, associated with spin formula_78 is more populated than the other. In the presence of a rotating magnetic field more protons flip from formula_118 to formula_81 than flip the other way, causing absorption of microwave or radio-wave radiation (from the rotating field) . When the field is withdrawn, protons tend to reequilibrate along the Boltzmann distribution, so some protons transition from higher energy level to lower ones, emitting microwave or radio-wave radiation at specific frequencies.
Instead of nuclear spin, spin angular momentum of unpaired electrons is used in EPR (Electron paramagnetic resonance) to detect free radicals, etc.
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.