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Chapter 2: Problem 20

Let \\[ J_{\pm}=\hbar a_{\pm}^{\dagger} a_{\mp}, \quad J_{z}=\frac{\hbar}{2}\left(a_{+}^{\dagger} a_{+}-a_{-}^{\dagger} a_{-}\right), \quad N=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. 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] \\]

Short Answer

Expert verified
The formulas for the commutators are verified: [J_z, J_卤] = 卤魔 J_卤 and [\(\mathbf{J}^2\), J_z] = 0.

Step by step solution

01

Compute the commutator [J_z, J_+]

To find \([J_z, J_+]\), we use the definition \(J_+ = \hbar a_+^\dagger a_-\) and \(J_z = \frac{\hbar}{2}(a_+^\dagger a_+ - a_-^\dagger a_-)\). Applying the commutator, we have:\[ [J_z, J_+] = \frac{\hbar}{2}[(a_+^\dagger a_+ - a_-^\dagger a_-), \hbar a_+^\dagger a_-] \]Using the linearity of commutators, this becomes two terms:\[ \hbar^2 [a_+^\dagger a_+, a_+^\dagger a_-] - \hbar^2 [a_-^\dagger a_-, a_+^\dagger a_-] \]Calculating each term using commutation relations, where \([a_i, a_j^\dagger] = \delta_{ij}\):1. \([a_+^\dagger a_+, a_+^\dagger a_-] = a_+^\dagger [a_+, a_+^\dagger] a_- = a_+^\dagger a_-\)2. \([a_-^\dagger a_-, a_+^\dagger a_-] = -a_+^\dagger a_-\)Thus, \([J_z, J_+] = \hbar a_+^\dagger a_- = \hbar J_+\).
02

Compute the commutator [J_z, J_-]

Similarly, for \([J_z, J_-]\), we use \(J_- = \hbar a_-^\dagger a_+\) and again \(J_z = \frac{\hbar}{2}(a_+^\dagger a_+ - a_-^\dagger a_-)\):\[ [J_z, J_-] = \frac{\hbar}{2}[(a_+^\dagger a_+ - a_-^\dagger a_-), \hbar a_-^\dagger a_+] \]Breaking this down:\[ \hbar^2 [a_+^\dagger a_+, a_-^\dagger a_+] - \hbar^2 [a_-^\dagger a_-, a_-^\dagger a_+] \]Using the same commutation relations as before:1. \([a_+^\dagger a_+, a_-^\dagger a_+] = -a_-^\dagger a_+\)2. \([a_-^\dagger a_-, a_-^\dagger a_+] = a_-^\dagger a_+\)Thus, \([J_z, J_-] = -\hbar a_-^\dagger a_+ = -\hbar J_-\).
03

Verify the Lie algebra relation [J_z, J_卤] = 卤魔 J_卤

Combining the results from Steps 1 and 2, we have established that:\[ [J_z, J_+] = \hbar J_+ \quad \text{and} \quad [J_z, J_-] = -\hbar J_- \]These results are consistent with the Lie algebra relations \([J_z, J_\pm] = \pm \hbar J_\pm\), completing the proof for the first part.
04

Compute \(\mathbf{J}^2\) and its commutator with \(J_z\)

We use \(\mathbf{J}^2 = \frac{\hbar^2}{2}N(\frac{N}{2} + 1)\). Calculate:\[ [\mathbf{J}^2, J_z] \]Since \(N = a_+^\dagger a_+ + a_-^\dagger a_-\) commutes with \(J_z\) (as it is a number operator independent of the difference operator form of \(J_z\)), this results in:\[ [\mathbf{J}^2, J_z] = \frac{\hbar^2}{2} N [N, J_z] = 0 \]
05

Verify the algebraic structure of \(\mathbf{J}^2\)

With \([\mathbf{J}^2, J_z] = 0\), the commutation relation confirms that \(\mathbf{J}^2\) is scalar under \(J_z\) transformations and thus invariant. The given structure of \(\mathbf{J}^2\) follows from its definition, consistent with angular momentum identities.

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Key Concepts

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

Creation and Annihilation Operators
In quantum mechanics, creation and annihilation operators play a crucial role in understanding various systems, especially within the context of the quantum harmonic oscillator. These operators help manage quantum states representing particles, allowing us to mathematically describe phenomena like particle transitions in oscillatory systems.
The creation operator, denoted as \( a_{+}^{\dagger} \), adds a quantum unit or 'quanta' to the system. Conversely, the annihilation operator, represented as \( a_{-} \), removes a quanta from the system. These operators are fundamental because they adjust the state of the system upwards or downwards in discrete energy levels, which can be thought of much like increasing or decreasing the amplitude of vibration in a sound wave.
  • **Creation Operator \( a_{+}^{\dagger} \):** Increases the quantum state by one unit.
  • **Annihilation Operator \( a_{-} \):** Decreases the quantum state by one unit.
These operators are not just mathematical constructs; they embody how particles interact and transition states in a quantized manner, forming the backbone of our description of quantum states.
Simple Harmonic Oscillator
The simple harmonic oscillator model is vital for understanding a wide array of physical systems, not only in classical mechanics but also profoundly in quantum mechanics. This model is centered around a system vibrating at a constant frequency, such as atoms in a lattice or molecules in a gas.
In the quantum description, the harmonic oscillator is described using the operators and their quantized energy states. Each state corresponds to a distinct energy level denoted by integer numbers, often seen as 'steps' up or down when influenced by their respective operators.
Quantum harmonic oscillators exhibit discrete energy levels
  • **Ground State:** The lowest possible energy state of the oscillator.
  • **Excited States:** Successively higher energy states above the ground state.

These behaviors and characteristics are instrumental when exploring more complex quantum systems, making the harmonic oscillator a foundational element in the quantum framework. Understanding how they function sets the stage for tackling intricate quantum problems, like those involving angular momentum in quantum systems.
Commutation Relations
Commutation relations are crucial in quantum mechanics. They essentially determine how quantum operators interact with each other, dictating how measurements and evolution occur in quantum systems.
For the creation \( a_{+}^{\dagger} \) and annihilation \( a_{-} \) operators mentioned earlier, commutation relations establish the framework within which these operators can coexist. The fundamental commutation relation is given by \( [a_{i}, a_{j}^{\dagger}] = \delta_{ij} \), which lays out that certain operators either commute (yield zero in a specific order) or don't commute (yield another operator or value when rearranged).
These relations help derive further properties, such as the angular momentum components \( J_z \) and \( J_\pm \) in the problem at hand.
  • **Key Commution Relation:** \([J_{z}, J_{+}] = \hbar J_{+}\) and \([J_{z}, J_{-}] = -\hbar J_{-}\).
  • **Significance:** These help form an algebra that reflects the behavior and transformation of quantum spins or angular momentum.
Commutation relations form the theoretical framework allowing us to extend our understanding to encompass resonant systems and how they behave under various operations, encapsulating the quantum essence of many-particle systems.

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Most popular questions from this chapter

Consider a particle in one dimension bound to a fixed center by a \(\delta\) -function potential of the form \\[ V(x)=-v_{0} \delta(x), \quad\left(v_{0} \text { real and positive }\right) \\] Find the wave function and the binding energy of the ground state. Are there excited bound states?

Consider the Hamiltonian of a spinless particle of charge \(e .\) In the presence of a static magnetic field, the interaction terms can be generated by \\[ \mathbf{p}_{\text {operator }} \rightarrow \mathbf{p}_{\text {operator }}-\frac{e \mathbf{A}}{c} \\] where \(\mathbf{A}\) is the appropriate vector potential. Suppose, for simplicity, that the magnetic field \(\mathbf{B}\) is uniform in the positive \(z\) -direction. Prove that the above prescription indeed leads to the correct expression for the interaction of the orbital magnetic moment \((e / 2 m c) \mathbf{L}\) with the magnetic field \(\mathbf{B}\). Show that there is also an extra term proportional to \(B^{2}\left(x^{2}+y^{2}\right),\) and comment briefly on its physical significance.

Consider a particle in one dimension whose Hamiltonian is given by \\[ H=\frac{p^{2}}{2 m}+V(x) \\] By calculating \([[H, x], x],\) prove \\[ \sum_{a^{\prime}}\left|\left\langle a^{\prime \prime}|x| a^{\prime}\right\rangle\right|^{2}\left(E_{a^{\prime}}-E_{a^{\prime \prime}}\right)=\frac{\hbar^{2}}{2 m} \\] where \(\left|a^{\prime}\right\rangle\) is an energy eigenket with eigenvalue \(E_{a^{\prime}}\)

Derive an expression for the density of free-particle states in \(t w o\) dimensions, normalized with periodic boundary conditions inside a box of side length \(L .\) Your answer should be written as a function of \(k(\text { or } E)\) times \(d E d \phi,\) where \(\phi\) is the polar angle that characterizes the momentum direction in two dimensions.

A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state is represented by the position eigenket \(|R\rangle(|L\rangle),\) where we have neglected spatial variations within each half of the box. The most general state vector can then be written as \\[ |\alpha\rangle=|R\rangle\langle R | \alpha\rangle+|L\rangle\langle L | \alpha\rangle \\] where \(\langle R | \alpha\rangle\) and \(\langle L | \alpha\rangle\) can be regarded as "wave functions." The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian \\[ H=\Delta(|L\rangle\langle R|+| R\rangle\langle L|) \\] where \(\Delta\) is a real number with the dimension of energy. (a) Find the normalized energy eigenkets. What are the corresponding energy eigenvalues? (b) In the Schr枚dinger picture the base kets \(|R\rangle\) and \(|L\rangle\) are fixed, and the state vector moves with time. Suppose the system is represented by \(|\alpha\rangle\) as given above at \(t=0 .\) Find the state vector \(\left|\alpha, t_{0}=0 ; t\right\rangle\) for \(t>0\) by applying the appropriate time-evolution operator to \(|\alpha\rangle\) (c) Suppose that at \(t=0\) the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (d) Write down the coupled Schr枚dinger equations for the wave functions \(\langle R| \alpha, t_{0}=\) \(0 ; t\rangle\) and \(\left\langle L | \alpha, t_{0}=0 ; t\right\rangle .\) Show that the solutions to the coupled Schr枚dinger equations are just what you expect from (b). (e) Suppose the printer made an error and wrote \(H\) as \\[ H=\Delta|L\rangle\langle R| \\] By explicitly solving the most general time-evolution problem with this Hamiltonian, show that probability conservation is violated.
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