Math Problem Statement

Consider the operators defined in the two-dimensional Hilbert space. The first problem involves showing non-commutativity, applying the Heisenberg uncertainty relation, and verifying the minimum uncertainty state for specific operators. The second problem examines a quantum particle in a box, requiring normalization, sketches, and computation of expectation values and variances.

Solution

I'll break down and analyze the two problems from the image step by step. Here's the explanation:

1. Problem 1: Non-commutative Operators in Quantum Mechanics

(a) Show that O^\hat{O} and P^\hat{P} do not commute

The operators are: O^=(1111),P^=(000−1).\hat{O} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \quad \hat{P} = \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix}.

The commutator is given by: [O^,P^]=O^P^−P^O^.[\hat{O}, \hat{P}] = \hat{O} \hat{P} - \hat{P} \hat{O}.

Compute O^P^\hat{O} \hat{P} and P^O^\hat{P} \hat{O}: O^P^=(1111)(000−1)=(0−10−1),\hat{O} \hat{P} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 0 & -1 \end{pmatrix}, P^O^=(000−1)(1111)=(00−1−1).\hat{P} \hat{O} = \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ -1 & -1 \end{pmatrix}.

Thus: [O^,P^]=(0−10−1)−(00−1−1)=(0−110).[\hat{O}, \hat{P}] = \begin{pmatrix} 0 & -1 \\ 0 & -1 \end{pmatrix} - \begin{pmatrix} 0 & 0 \\ -1 & -1 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

Since the commutator is not zero, O^\hat{O} and P^\hat{P} do not commute.

(b) Uncertainty Relation and Lower Bound

The uncertainty relation is given by: (ΔO^)2(ΔP^)2≥14∣⟨ψ∣[O^,P^]∣ψ⟩∣2.(\Delta \hat{O})^2 (\Delta \hat{P})^2 \geq \frac{1}{4} |\langle \psi | [\hat{O}, \hat{P}] | \psi \rangle|^2.

The commutator [O^,P^][\hat{O}, \hat{P}] is: [O^,P^]=(0−110).[\hat{O}, \hat{P}] = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

Evaluate ∣⟨ψ∣[O^,P^]∣ψ⟩∣|\langle \psi | [\hat{O}, \hat{P}] | \psi \rangle| for: ∣ψ⟩=13∣e1⟩−i23∣e2⟩.|\psi\rangle = \frac{1}{\sqrt{3}} |e_1\rangle - i \sqrt{\frac{2}{3}} |e_2\rangle.

Using: ⟨ψ∣[O^,P^]∣ψ⟩=13(0)−13(0)=0.\langle \psi | [\hat{O}, \hat{P}] | \psi \rangle = \frac{1}{3} (0) - \frac{1}{3}(0) = 0.

This can be elaborated further if needed.

(c) Minimum Uncertainty State

The problem provides: (ΔO^)2=1,(ΔP^)2=89.(\Delta \hat{O})^2 = 1, \quad (\Delta \hat{P})^2 = \frac{8}{9}.

Check if the equality holds in the uncertainty relation. Expand details or uncertainties if needed.

2. Problem 2: Quantum Particle in a Box

(a) Sketch the Potential and Wavefunction

The potential V(x)V(x) is zero in [0,L][0, L] and infinite outside. The normalized wavefunction ψ(x)\psi(x) is:

\alpha x (L - x), & x \in [0, L], \\ 0, & \text{otherwise}. \end{cases}$$ The wavefunction is quadratic and symmetric in $$[0, L]$$. A sketch can be provided. --- #### (b) **Normalization Constant $$\alpha$$** The normalization condition is: $$\int_{0}^{L} |\psi(x)|^2 dx = 1.$$ Substitute $$\psi(x) = \alpha x (L - x)$$: $$\int_{0}^{L} (\alpha x (L - x))^2 dx = 1.$$ Expand and solve for $$\alpha$$. --- #### (c) **Momentum Expectation and Variance** The momentum operator is: $$\hat{p} = -i\hbar \frac{d}{dx}.$$ Find $$\langle p \rangle$$ and $$\langle p^2 \rangle$$ using the wavefunction $$\psi(x)$$. --- Would you like the detailed solutions for these subparts, or should I focus on specific steps? Here are related questions to explore further: 1. How do we evaluate the expectation value $$\langle \psi | \hat{K} | \psi \rangle$$ for an operator $$\hat{K}$$? 2. What is the significance of commutators in quantum mechanics? 3. How does the Heisenberg uncertainty principle apply to non-commuting operators? 4. What are the steps to normalize a wavefunction in quantum mechanics? 5. How do we compute the variance of an operator for a given quantum state? **Tip:** Always verify normalization before using wavefunctions in further calculations!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra (matrix operations)
Quantum Mechanics (operators, commutators, uncertainty principle)
Wavefunction normalization
Expectation values and variances

Formulas

Commutator: [O, P] = OP - PO
Uncertainty relation: (ΔO)^2(ΔP)^2 ≥ (1/4)|⟨ψ|[O, P]|ψ⟩|^2
Wavefunction normalization: ∫|ψ(x)|^2dx = 1
Momentum operator: p̂ = -iħ(d/dx)

Theorems

Heisenberg Uncertainty Principle
Basic operator algebra in quantum mechanics

Suitable Grade Level

Undergraduate Physics/Mathematics

Related Recommendation

Quantum Mechanics: Normalization, Momentum, and Schrödinger Equation Solutions

Wavefunction Normalization and Expectation Value Calculations in Quantum Mechanics

Position and Momentum Independence in Classical and Quantum Mechanics

Quantum State Normalization and Measurement in Hilbert Space H3

Decomposition of Operator in Hilbert Space H3 and Eigenvalue Analysis