Math Problem Statement
Suppose a vector in H1 is |v(1)⟩ = 1/√2 |1(1)⟩ + I/√2 |2(1)⟩ while a vector in H2 is |v(2)⟩ = 1/√3 |1(2)⟩ + √2/√3 |2(2)⟩. Show that each of these vectors are normalized.
Solution
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Math Problem Analysis
Mathematical Concepts
Quantum Mechanics
Vector Spaces
Linear Algebra
Hilbert Spaces
Formulas
⟨v | v⟩ = 1 (normalization condition)
|v(1)⟩ = 1/√2 |1(1)⟩ + i/√2 |2(1)⟩
|v(2)⟩ = 1/√3 |1(2)⟩ + √2/√3 |2(2)⟩
⟨v(1) | v(1)⟩ = ⟨v(2) | v(2)⟩ = 1 (inner product calculation)
Theorems
Normalization of vectors
Orthonormality of basis vectors in Hilbert space
Suitable Grade Level
Undergraduate (Quantum Mechanics or Linear Algebra)
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