Math Problem Statement

please solve 3a and b

Solve 3a and b from the image: 3a asks for the allowed values for the quantum number m when ℓ = 1, and 3b requires the normalized wavefunction that is an equal superposition of all eigenstates.

Solution

Question 3a: Allowed values for mm

In a system with total angular momentum quantum number =1\ell = 1, the magnetic quantum number mm can take on the following values:

m=,+1,,1,m = -\ell, -\ell + 1, \dots, \ell - 1, \ell

Substituting =1\ell = 1: m=1,0,1m = -1, 0, 1

Thus, the allowed values for mm are: m=1,0,1m = -1, 0, 1


Question 3b: Normalized wavefunction ψ|\psi\rangle

The problem asks for a normalized wavefunction ψ|\psi\rangle that is an equal superposition of all possible eigenstates =1,m| \ell = 1, m \rangle, where mm takes the values 1,0,1-1, 0, 1.

A general superposition of these eigenstates can be written as: ψ=a1,1+b1,0+c1,1|\psi\rangle = a |1, -1\rangle + b |1, 0\rangle + c |1, 1\rangle

To ensure the wavefunction is an equal superposition, the coefficients aa, bb, and cc must have equal magnitudes. Therefore, each coefficient is chosen to be 13\frac{1}{\sqrt{3}}, ensuring normalization.

The normalized wavefunction becomes: ψ=131,1+131,0+131,1|\psi\rangle = \frac{1}{\sqrt{3}} |1, -1\rangle + \frac{1}{\sqrt{3}} |1, 0\rangle + \frac{1}{\sqrt{3}} |1, 1\rangle

Alternatively, in summation form: ψ=13m=111,m|\psi\rangle = \frac{1}{\sqrt{3}} \sum_{m=-1}^{1} |1, m\rangle


Would you like further explanation of these results or additional details?

Follow-up Questions:

  1. What are the physical meanings of the quantum numbers \ell and mm?
  2. How does normalization ensure the total probability equals 1?
  3. Why are the coefficients in part (b) all equal for an equal superposition?
  4. What would change in the wavefunction if \ell were a different value?
  5. How does the summation notation simplify writing superpositions?

Tip:

Always verify normalization by calculating the total probability ψψ\langle \psi|\psi \rangle to ensure it equals 1.

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Math Problem Analysis

Mathematical Concepts

Quantum Mechanics
Angular Momentum
Superposition Principle
Normalization

Formulas

m = -ℓ, -ℓ + 1, ..., ℓ
Normalization: |ψ⟩ = Σ c_n |n⟩ with Σ |c_n|^2 = 1

Theorems

Quantum Angular Momentum Rules
Normalization Theorem

Suitable Grade Level

Undergraduate Physics (Advanced Level)