In , with the standard inner product , define the three column vectors |1 = 1 3…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Hilbert space and Dirac notation

Problem

In C3\mathbb{C}^3, with the standard inner product ⟨ϕ∣ψ⟩=∑iϕi∗ψi\langle\phi|\psi\rangle = \sum_i \phi_i^*\psi_i, define the three column vectors∣1⟩=13(111),∣2⟩=12(1−10),∣3⟩=16(11−2)|1\rangle = \frac{1}{\sqrt3}\begin{pmatrix}1\\1\\1\end{pmatrix},\quad |2\rangle = \frac{1}{\sqrt2}\begin{pmatrix}1\\-1\\0\end{pmatrix},\quad |3\rangle = \frac{1}{\sqrt6}\begin{pmatrix}1\\1\\-2\end{pmatrix} (a) Verify that {∣1⟩,∣2⟩,∣3⟩}\{|1\rangle,|2\rangle,|3\rangle\} is an orthonormal basis of C3\mathbb{C}^3. (b) The normalised state ∣ψ⟩=15(2,0,1)T|\psi\rangle = \frac{1}{\sqrt5}(2,0,1)^T is given. Find the components cn=⟨n∣ψ⟩c_n = \langle n|\psi\rangle for n=1,2,3n=1,2,3. (c) Check Parseval's identity ∑n∣cn∣2=⟨ψ∣ψ⟩\sum_n|c_n|^2 = \langle\psi|\psi\rangle.

Related theory

Browse the problem library