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Chapter 2: Problem 50

Consider the neutron interferometer. Prove that the difference in the magnetic fields that produce two successive maxima in the counting rates is given by $$ B=\frac{4 \pi \hbar c}{|e| g_{n} \hbar l} $$ where \(g_{n}(=-1.91)\) is the neutron magnetic moment in units of \(-e \hbar / 2 m_{n} c\), and \(\lambda \equiv\) \(\lambda / 2 \pi\). This problem was in fact analyzed in the paper by Bernstein, Phys. Rev. Lett., 18 (1967) 1102.

Short Answer

Expert verified
The difference in magnetic fields for successive maxima is \( B = \frac{4\pi \hbar c}{|e| g_n l} \).

Step by step solution

01

Understanding Variables and Constants

The exercise involves finding the difference in magnetic fields that result in successive maxima in an interferometer using given constants and relationships. Here, - \( |e| \) is the absolute value of the electron charge,- \( g_n \) is the neutron magnetic moment,- \( \hbar \) is the reduced Planck's constant,- \( c \) is the speed of light,- \( l \) is the path length.
02

Recall Neutron Interference Condition

Successive maxima in the interference pattern occur when the phase difference \( \Delta \phi \) between the paths differs by \( 2\pi n \) (where \( n \) is an integer). The phase difference caused by a magnetic field \( B \) is given by \[ \Delta \phi = \dfrac{2 |e|}{\hbar c m_n} g_n B l \]
03

Calculate Change Required for Successive Maxima

For successive maxima, the change in phase difference \( \Delta \phi \) should be \( 2\pi \). Set the phase difference expression to equal \( 2\pi \) and solve for \( B \) using \( n = 1 \).\[ \frac{2 |e|}{\hbar c m_n} g_n B l = 2\pi \]
04

Rearrange Equation for B

Rearrange the equation to solve for \( B \):\[ B = \frac{4\pi \hbar c}{|e| g_n l} \]
05

Verify Constants and Units

Check that all constants and units are correctly accounted for, ensuring that the dimensions on both sides of the equation match to confirm the correctness of the result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Fields
Magnetic fields are invisible forces that can exert a force on certain particles, like neutrons, if they possess a magnetic moment. The field is represented by the symbol \( B \) and its unit is the Tesla (T). In the context of a neutron interferometer, the magnetic fields influence how neutrons travel through different paths. This is crucial in the experiment to create interference patterns and locate points of constructive interference, where maxima occur.
  • Magnitude and Direction of Magnetic Fields: The magnitude is measured in Tesla while direction is defined by lines of flux.
  • Impact on Neutrons: Neutrons, having a magnetic moment, interact with these fields, altering their path and the interference pattern observed.
Understanding magnetic fields and their interaction with neutron magnetic moments is key for predicting and analyzing results in experiments involving neutron interferometers.
Phase Difference
Phase difference, denoted by \( \Delta \phi \), is a vital concept in interference phenomena, particularly in a neutron interferometer. This difference in phase affects whether neutrons interfere constructively or destructively when they reunite.
  • What is Phase Difference? It's the disparity in the phase of waves traveling different paths; in this case, neutron paths in the interferometer.
  • Role in Interference: When neutrons experience a phase difference of \( 2\pi \), they reach a point of constructive interference, creating maxima.
  • Calculation: The path length and magnetic field both contribute to changing the phase as per the formula \( \Delta \phi = \dfrac{2 |e|}{\hbar c m_n} g_n B l \).
This principle governs how we calculate the necessary magnetic field conditions for successive maxima in the interference pattern.
Neutron Magnetic Moment
Neutrons have a property called the magnetic moment, represented by \( g_n \). Despite being electrically neutral, neutrons have a magnetic moment due to their internal quark structure and the movement of charges within them.
  • Significance: It measures how a neutron interacts with external magnetic fields, which can influence their trajectory in a neutron interferometer.
  • Given as a Constant: In the problem, \( g_n = -1.91 \), a dimensionless quantity, notable for its negative sign indicating the orientation of the neutron's magnetic moment opposite to the field.
  • Role in Calculations: The magnetic moment is part of the phase difference equation, determining how changes in magnetic fields alter the interference pattern.
Understanding the neutron magnetic moment helps in comprehending the mechanics of interference patterns observed in neutron interferometry.
Interference Pattern
When neutrons pass through a neutron interferometer, they create an interference pattern. This pattern results from the superposition of waves that have traveled through different paths, influenced by the magnetic field, length of the paths, and the magnetic moment of the neutron.
  • Constructive vs Destructive Interference: Constructive interference occurs at points where path differences equal multiples of \( 2\pi \), producing bright spots or maxima, while destructive interference results in dark spots or minima.
  • Dependence on Phase Difference: The phase difference influences the type and location of interference, dictating whether the patterns are bright or dark.
  • Applications: By observing changes in interference patterns, scientists can infer the properties of particles and fields impacting the neutrons.
Analyzing these patterns allows scientists to measure subatomic influences and confirm theoretical predictions, making them a cornerstone concept in quantum mechanics.

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Most popular questions from this chapter

A particle in one dimension is trapped between two rigid walls: $$ V(x)=\left\\{\begin{array}{lll} 0 & \text { for } & 0L . \end{array}\right. $$ At \(t=0\) it is known to be exactly at \(x=L / 2\) with certainty. What are the relative probabilities for the particle to be found in various energy eigenstates? Write down the wave function for \(t \geq 0\). (You need not worry about absolute normalization, convergence, and other mathematical subtleties.)

A particle in one dimension \((-\infty0) . $$ a. Is the energy spectrum continuous or discrete? Write down an approximate expression for the energy eigenfunction specified by \(E\). Also sketch it crudely. b. Discuss briefly what changes are needed if \(V\) is replaced by $$ V=\lambda|x| . $$

Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose at \(t=0\) the state vector is given by $$ \exp \left(\frac{-i p a}{\hbar}\right)|0\rangle, $$ where \(p\) is the momentum operator, \(a\) is some number with dimension of length, and the state \(|0\rangle\) is the one for which \(\langle x\rangle=0=\langle p\rangle\). Using the Heisenberg picture, evaluate the expectation value \(\langle x\rangle\) for \(t \geq 0\).

Show for the one-dimensional simple harmonic oscillator $$ \left\langle 0\left|e^{i k x}\right| 0\right\rangle=\exp \left[-k^{2}\left\langle 0\left|x^{2}\right| 0\right\rangle / 2\right], $$ where \(x\) is the position operator.

Make the definitions $$ J_{\pm} \equiv \hbar a_{\pm}^{\dagger} a_{\mp}, \quad J_{z} \equiv \frac{\hbar}{2}\left(a_{+}^{\dagger} a_{+}-a_{-}^{\dagger} a_{-}\right), \quad N \equiv a_{+}^{\dagger} a_{+}+a_{-}^{\dagger} a_{-} $$ where \(a_{\pm}\)and \(a_{\pm}^{\dagger}\) are the annihilation and creation operators of two independent simple harmonic oscillators satisfying the usual simple harmonic oscillator commutation relations. Also make the definition $$ \mathbf{J}^{2} \equiv J_{z}^{2}+\frac{1}{2}\left(J_{+} J_{-}+J_{-} J_{+}\right) $$ Prove $$ \left[J_{z}, J_{\pm}\right]=\pm \hbar J_{\pm}, \quad\left[\mathbf{J}^{2}, J_{z}\right]=0, \quad \mathbf{J}^{2}=\left(\frac{\hbar^{2}}{2}\right) N\left[\left(\frac{N}{2}\right)+1\right] $$
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