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Chapter 1: Problem 32

The translation operator for a finite (spatial) displacement is given by $$ \mathscr{J}(\mathbf{l})=\exp \left(\frac{-i \mathbf{p} \cdot \mathbf{l}}{\hbar}\right) $$ where \(\mathbf{p}\) is the momentum operator. a. Evaluate $$ \left[x_{i}, \mathscr{J}(\mathbf{l})\right] . $$ b. Using (a) (or otherwise), demonstrate how the expectation value \(\langle\mathbf{x}\rangle\) changes under translation.

Short Answer

Expert verified
a. \([x_i, \mathscr{J}(\mathbf{l})] = l_i \mathscr{J}(\mathbf{l})\). b. The expectation value \(\langle \mathbf{x} \rangle\) shifts by \(\mathbf{l}\).

Step by step solution

01

Understand the Commutation Relation

To evaluate \([x_i, \mathscr{J}(\mathbf{l})]\), we start with the fundamental commutation being sought: \([x_i, \mathscr{J}(\mathbf{l})]\). We have the translation operator \(\mathscr{J}(\mathbf{l}) = \exp\left(\frac{-i \mathbf{p} \cdot \mathbf{l}}{\hbar}\right)\). We represent the momentum operator \(\mathbf{p}\) as \(-i\hbar abla\).
02

Recognize Operator Exponential Property

We use the property of exponentials involving operators, specifically that \(e^{A} B e^{-A} = B + [A, B] + \frac{1}{2!} [A, [A, B]] + \ldots\). Hence, we take \(A = \frac{-i \mathbf{p} \cdot \mathbf{l}}{\hbar}\) and \(B = x_i\).
03

Derive Commutator Expression

Applying \([\mathbf{p} \cdot \mathbf{l}, x_i]\), we have \([p_j l_j, x_i] = [p_i l_i, x_i] = i \hbar l_i\) using \([p_i, x_j] = -i \hbar \delta_{ij}\). The remaining terms vanish due to symmetries and operator properties, simplifying the expression.
04

Simplify Commutator

Accounting for only the non-zero commutator contributions, we obtain \([x_i, \mathscr{J}(\mathbf{l})] = \mathscr{J}(\mathbf{l}) l_i\), suggesting \([x_i, \mathscr{J}(\mathbf{l})] = l_i \mathscr{J}(\mathbf{l})\) as the final result of part (a).
05

Analyze Expectation Value Change

Given \( \langle \mathbf{x} \rangle \) changes under \( \mathscr{J}(\mathbf{l}) \), we find \(\langle \psi | x_i | \psi \rangle \rightarrow \langle \psi | x_i + l_i | \psi \rangle\). By linearity, this represents a shift by \( \mathbf{l} \) in expectation: \( \langle \mathbf{x} \rangle' = \langle \mathbf{x} \rangle + \mathbf{l}\).

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Key Concepts

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

Translation Operator
In quantum mechanics, the translation operator is a fundamental concept that shifts a quantum state in space. It's defined as \( \mathscr{J}(\mathbf{l}) = \exp \left(\frac{-i \mathbf{p} \cdot \mathbf{l}}{\hbar}\right) \). This operator acts on the wave function to move it a distance \( \mathbf{l} \).
\( \ \)
  • Here, \( \mathbf{p} \) is the momentum operator, a key player in how states evolve in space.
  • The factor \( \hbar \) is the reduced Planck's constant, necessary for the relation of energy to frequency and momentum to wavelength.
  • The exponential function hints at the rotational nature of quantum translations, similar to phase changes in waves.
Translation operators are particularly useful in studying the effects of symmetry in physical systems, as they demonstrate how wave functions and observable quantities transform when the system's origin is changed.
Momentum Operator
The momentum operator \( \mathbf{p} \) is vital in quantum mechanics because it connects position and translational motion in state spaces. Represented as \( \mathbf{p} = -i \hbar abla \), this operator captures how much and in which direction a wave function is 'flowing.'
\( \ \)
  • The minus sign and the imaginary unit \( i \) indicate a phase shift in the wave function, demonstrating its wave-like nature.
  • \( abla \) is the gradient operator, symbolizing the presence of change or shift in space.
  • Momentum operators are particularly linked with translations; their commutator with position reveals how states evolve when position is changed.
In practical applications, the momentum operator tells us how particles 'move' through space and how their paths are defined in quantum terms. It鈥檚 crucial for understanding particle behaviors and energies when applied to wave functions.
Expectation Value
In quantum mechanics, the expectation value \( \langle \mathbf{x} \rangle \) of an observable represents what we expect to measure on average if we perform many measurements of the system. It鈥檚 a statistical mean derived from probabilistic interpretations.
\( \ \)
  • For a position \( x \), the expectation value \( \langle x \rangle \) is computed as \( \langle \psi | x | \psi \rangle \), indicating the average location of particles described by the wave function \( \psi \).
  • When a translation is applied, the expectation value shifts by the vector \( \mathbf{l} \), effectively indicating the new "average" position: \( \langle x \rangle' = \langle x \rangle + l \).
  • This shift mirrors the action of the translation operator, showing spatial displacement and confirming the operator's effect.
Expectancy values are crucial for quantum experiments as they predict outcomes and confirm quantum state models, correlating theoretical predictions with experimental results.
Commutation Relation
A commutation relation in quantum mechanics describes how two operators relate to each other when applied in sequence. For position \( x_i \) and the translation operator \( \mathscr{J}(\mathbf{l}) \), we analyze their commutation: \( [x_i, \mathscr{J}(\mathbf{l})] \).
\( \ \)
  • In this context, the fundamental relation \( [x_i, p_j] = -i \hbar \delta_{ij} \) is used, where \( \delta_{ij} \) is Kronecker delta.
  • Upon evaluating, the result \( [x_i, \mathscr{J}(\mathbf{l})] = l_i \mathscr{J}(\mathbf{l}) \) illustrates that commutation leads to a term proportional to the displacement \( \mathbf{l} \).
  • This specific commutation signifies how the translation operator modifies positions, introducing an intuitive sense of how much position shifts due to momentum-induced translation.
Understanding these relations allows us to predict how quantum systems behave under different operations and is essential for mathematical formulations within quantum mechanics, assuring consistency and revealing foundational properties of operators.

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Most popular questions from this chapter

Two Hermitian operators anticommute: $$ \\{A, B\\}=A B+B A=0 . $$ Is it possible to have a simultaneous (that is, common) eigenket of \(A\) and \(B ?\) Prove or illustrate your assertion.

A beam of spin \(\frac{1}{2}\) atoms goes through a series of Stern-Gerlach type measurements as follows. a. The first measurement accepts \(s_{z}=\hbar / 2\) atoms and rejects \(s_{z}=-\hbar / 2\) atoms. b. The second measurement accepts \(s_{n}=\hbar / 2\) atoms and rejects \(s_{n}=-\hbar / 2\) atoms, where \(s_{n}\) is the eigenvalue of the operator \(\mathbf{S} \cdot \hat{\mathbf{n}}\), with \(\hat{\mathbf{n}}\) making an angle \(\beta\) in the \(x z\)-plane with respect to the \(z\)-axis. c. The third measurement accepts \(s_{z}=-\hbar / 2\) atoms and rejects \(s_{z}=\hbar / 2\) atoms. What is the intensity of the final \(s_{z}=-\hbar / 2\) beam when the \(s_{z}=\hbar / 2\) beam surviving the first measurement is normalized to unity? How must we orient the second measuring apparatus if we are to maximize the intensity of the final \(s_{z}=-\hbar / 2\) beam?

A two-state system is characterized by the Hamiltonian $$ H=H_{11}|1\rangle\left\langle 1\left|+H_{22}\right| 2\right\rangle\langle 2|+H_{12}[|1\rangle\langle 2|+| 2\rangle\langle 1|] $$ where \(H_{11}, H_{22}\), and \(H_{12}\) are real numbers with the dimension of energy, and \(|1\rangle\) and \(|2\rangle\) are eigenkets of some observable \((\neq H)\). Find the energy eigenkets and corresponding energy eigenvalues. Make sure that your answer makes good sense for \(H_{12}=0\).

The Hamiltonian operator for a two-state system is given by $$ H=a(|1\rangle\langle 1|-| 2\rangle\langle 2|+| 1\rangle\langle 2|+| 2\rangle\langle 1|) $$ where \(a\) is a number with the dimension of energy. Find the energy eigenvalues and the corresponding energy eigenkets (as linear combinations of \(|1\rangle\) and \(|2\rangle\) ).

A certain observable in quantum mechanics has a \(3 \times 3\) matrix representation as follows: $$ \frac{1}{\sqrt{2}}\left(\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right) $$ a. Find the normalized eigenvectors of this observable and the corresponding eigenvalues. Is there any degeneracy? b. Give a physical example where all this is relevant.
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