Magnetic Resonance

We consider an electron with only spin degree of freedom in a region where a large static magnetic field Bo is along the z axis and a small oscillating field B_1 cos wt is along the x axis.

We know hamiltonian operator

The solution is in the form

Substituting the proposed solution in the above equation, we get,

Similarly from the second equation, we obtain

It is due to the fact that the term e^i(w + w_o)t oscillates much more repidly than the term e^i(2 w_o - w)t so on average, the first term gives zero contribution. Then we can write

Similarly we can say

Taking the derivative of the first and then putting for Ȧ_(t) from the second and A_(t) from the first we obtain

We write the solution of this equation in the form

General solution is

From equation 1,

Also,

Let the spin be up, pointing in the positive z direction at t = 0. This indicates that

The probability that at time t the spin is reversed (spin down state) is given by

Probability of finding the particles in spin down is small. But when w approaches to 2wo.

l C_(t) l ² = 1 - cos w_1 t / 2

At certain point,

l C_(t) l ² = 1

This equation holds significance in various areas of physics, particularly in quantum mechanics and spectroscopy, where it often arises in the context of wavefunctions and probability amplitudes.

Here's what it might signify in the context of magnetic resonance:

1. Normalization: In quantum mechanics, wavefunctions must be normalized, meaning that the integral of the squared magnitude of the wavefunction over all space must equal 1. This ensures that the probability of finding the particle somewhere in space is 100%. The equation you provided could be a manifestation of this normalization condition for a specific coefficient C(t) involved in a magnetic resonance phenomenon.

2. Conservation of Probability: In the context of magnetic resonance, the equation may represent the conservation of probability amplitude. In processes like nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI), where the behavior of atomic nuclei in a magnetic field is studied, the total probability amplitude should be conserved over time. The squared magnitude of C(t) represents the probability density, so the equation ensures that the total probability remains constant.

3. Signal Intensity: In spectroscopy, particularly in nuclear magnetic resonance (NMR) spectroscopy, the intensity of spectral lines is proportional to the square of the amplitude of the corresponding wavefunction. Therefore, |C(t)|² = 1 could signify that the spectral intensity is normalized, ensuring that the total intensity of all spectral lines equals the total intensity of the spectrum.

4. Physical Constraints: In some cases, the equation might represent a physical constraint or property of the system under consideration. For instance, in certain quantum mechanical systems, the coefficients of a wavefunction might be subject to constraints that result in |C(t)|² = 1.

In any case, the specific interpretation would depend on the details of the system or phenomenon being studied in the context of magnetic resonance.

Probability of finding a particle in spin down step is equal to 1. We see that spin reversed place. The phenomenon that spin reversal ( rotation of the spin from the up state to down state). When the frequency of applied periodic magnetic field becomes equal to processional frequency is known as magnetic resonance.

This note is a part of the Physics Repository.