Math Problem Statement
please solve 3a and b
Solution
Question 3a: Allowed values for
In a system with total angular momentum quantum number , the magnetic quantum number can take on the following values:
Substituting :
Thus, the allowed values for are:
Question 3b: Normalized wavefunction
The problem asks for a normalized wavefunction that is an equal superposition of all possible eigenstates , where takes the values .
A general superposition of these eigenstates can be written as:
To ensure the wavefunction is an equal superposition, the coefficients , , and must have equal magnitudes. Therefore, each coefficient is chosen to be , ensuring normalization.
The normalized wavefunction becomes:
Alternatively, in summation form:
Would you like further explanation of these results or additional details?
Follow-up Questions:
- What are the physical meanings of the quantum numbers and ?
- How does normalization ensure the total probability equals 1?
- Why are the coefficients in part (b) all equal for an equal superposition?
- What would change in the wavefunction if were a different value?
- How does the summation notation simplify writing superpositions?
Tip:
Always verify normalization by calculating the total probability 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)
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