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Chapter 8: Problem 1

A dipole withour angular momenturn can simply rotate to align with the field (though i would oscillate uniess it could shed energy). One with angular momentum cannot. Why?

Short Answer

Expert verified
A dipole with no angular momentum can simply align itself with the field because it reaches a state of minimum energy this way. One with angular momentum will continue to rotate, despite the presence of the field, due to the conservation of angular momentum. The continued rotation means that the dipole will precess around the field direction rather than aligning with it.

Step by step solution

01

Understanding Dipoles and Magnetic Fields

A dipole in a magnetic field will always try to align itself with the field. This is because when the dipole aligns with the field, the system reaches a state of minimum potential energy. This principle applies to both an electric dipole in an electric field and a magnetic dipole in a magnetic field.
02

The Role of Angular Momentum

Angular momentum is a measure of the amount of rotation an object has. It is a conserved quantity, meaning it will not change unless an external torque acts on it. In the context of a dipole in a magnetic field, it means that if a dipole had angular momentum before entering the field, it will keep spinning even after it enters the field.
03

Energy Dissipation

Now imagine a dipole with no angular momentum. It could simply align with the magnetic field, reaching a state of minimum energy. If any oscillations occur during this process,friction-like effects could dissipate energy, and the dipole would eventually come to rest, perfectly aligned with the field.
04

Dipoles with Angular Momentum

However, if the dipole had nonzero angular momentum, things are different. Because of the conservation of angular momentum, the dipole cannot simply stop rotating and align with the field. This continuous rotation means that the angle between the dipole and the magnetic field periodically changes, causing the dipole to precess around the direction of the magnetic field.

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Key Concepts

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

Angular Momentum
Angular momentum is a fundamental concept that describes the rotation of an object. Think of it like an object's tendency to keep spinning. For a dipole, having angular momentum means it's already in motion, spinning around a certain axis. This is a key factor because angular momentum is a conserved quantity, meaning it doesn't just disappear.
  • If an object has angular momentum, it will continue spinning at the same rate and direction unless acted upon by an outside force.
  • In a magnetic field, a dipole with angular momentum can't just stop and line up with the field instantly, it keeps on spinning.
This is important because even in a new environment鈥攍ike entering a magnetic field鈥攖he dipole's inherent spin won't just vanish. The dipole's rotation affects how it interacts with the magnetic field and prevents it from simply aligning without any precession or change to its angular state.
Magnetic Field
A magnetic field is a region where magnetic forces can be experienced by objects with magnetic properties. It's like an invisible web pulling or orienting magnets and magnetic dipoles towards itself. In the context of this exercise, when a magnetic dipole enters a magnetic field, it seeks to align itself in a particular way鈥攖he direction that minimizes potential energy.
  • This alignment is because a system always "prefers" to be in a state of lower energy. Think of it like water flowing to the lowest point; energy systems do something similar.
  • A dipole without angular momentum will naturally turn and face the direction of the magnetic field lines.
However, once there's angular momentum at play, this natural alignment doesn't occur straightforwardly. The magnetic field tries to influence the dipole, but the conservation of angular momentum causes more complexity in this interaction, causing it to move in a more complex manner such as precession.
Energy Dissipation
Energy dissipation refers to the process by which a system loses energy over time, usually as heat or thru friction-like effects. For a dipole, once it starts aligning with the magnetic field, if it has no angular momentum, energy can naturally dissipate and allow a perfect alignment.
  • Without angular momentum, a dipole can slowly shed energy, becoming progressively more aligned until it rests in this state.
  • An oscillating dipole can lose energy via processes similar to friction, slowing down any wobbling or movement.
However, for a dipole with angular momentum, continuous rotation prevents it from reaching full alignment. Instead of stopping, it precesses鈥攁 behavior similar to how a spinning top wobbles. As a result, the energy dissipation isn't enough to stop its motion entirely unless external forces are applied to counteract the rotating momentum.

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Most popular questions from this chapter

Slater Determinant: A convenient and compact way of expressing multiparticle states of antisymmetric character for many fermions is the Slater determinant. lt is based on the fact that for \(N\) fermions there must be \(N\) different individual-particle states, or sets of quantum numbers. The ith state has sparial quantum numbers (which might be \(n_{i}, \ell_{i},\) and \(m_{c i}\) ) represented simply by \(n_{t}\) and spin quanturn number \(m_{s i^{i}}\). Were it occupied by the ith particle, the state would be \(\psi_{n}\left(x_{j}\right) m_{s i}\). A column corresponds to a given state and a row to a given particle. For instance, the first column corresponds to individualparticle state \(\psi_{n}(x,) m_{3},\) where \(j\) progresses (through the rows) from particle 1 to particle \(N\). The first row corresponds to particle I. which successively occupies all individual-particle states (progressing through the columns). (a) What property of determinants ensures that the multiparticle state is 0 if any two individualparticle states are identical? (b) What property of deterninants ensures that switching the labels on any two particles switches the sign of the multiparticle state?

Show that the frequency at which an electron's intrinsic magnetic dipole moment would process in a magnetic field is given by \(\omega \cong e B / m_{c}\). Calculate this frequency for a field of \(10 \mathrm{~T}\)

The electron is known to have a radius no larger than \(10^{-18} \mathrm{~m}\) If actually produced by circulating mass, its intrinsic angular momentum of roughly \(\hbar\) would imply very high speed. even if all that mass were as far from the axis as possible. (a) Using simply \(r p\) (from \(|r \times p|\) ) for the angular momentum of a mass at radius \(r\). obtain a rough value of \(p\) and show that it would imply highly relativistic speed. (b) At such speeds, \(E=y m c^{2}\) and \(p=\gamma\) mu combine to give \(E \cong p c\) (just as for the speedy photon). How does this energy compare with the known intenial energy of the electron?

The Zeeman effect occurs in sodium just as in hydrogen - sodium's lone 3 s valence electron behaves much as hydrogen's 1.5. Suppose sodium atoms are immersed in a \(0.1 \mathrm{~T}\) magnetic field. (a) Into how many levels is the \(3 p_{1 / 2}\) level split? (b) Determine the energy spacing between these states. (c) Into how many lines is the \(3 p_{1 / 2}\) to \(3 s_{1 / 2}\) spectral line split by the field? (d) Describe quantitatively the spacing of these lines. (e) The sodium doublet \((589.0 \mathrm{nm}\) and \(589.6 \mathrm{nm}\) ) is two spectral lines. \(3 p_{3 n} \rightarrow 3 s_{1 / 2}\) and \(3 p_{1 / 2} \rightarrow 3 s_{1 / 2}\) which are split according to the two different possible spin-orbit ener gies in the \(3 p\) state (see Exercise 60 ). Detemine the splitting of the sodium doublet (the energy diff erence between the two photons). How does it compare with the line splitting of part (d), and why?

The wave functions for the ground and first excited states of a simple hartnonic oscillator are \(A e^{-b x^{2} / 2}\) and B.xe \(^{-b x^{2} / 2}\). Suppose you have two particles occupying these two states. (a) If distinguishable, an acceptable wave function would be \(A e^{-b x_{1}^{2} / 2} B x_{2} e^{-b x_{2}^{2} / 2}\). Calculate the probability that both particles would be on the positive side of the origin and divide by the total probability for both being found over all values of \(x_{1}\) and. \(x\), (This kind of nonnalizing-as-we-go will streamline things.) (b) Suppose now that the particles are indistin. guishable. Using the \(\pm\) symbol to reduce your work. calculate the same probability ratio, but assuming that their multiparticle wave function is either symmetric on antisymmetric. Comment on your results.
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