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Texts: Unitary operators, Bloch sphere. In this problem, H = C^2. |1⟩, |2⟩ are the basis vectors. (1) Note that |1⟩, |2⟩ are equal to the eigenvectors of H. (a) Show that a linear operator of the form |1⟩⟨2| + |2⟩⟨1| (2) is unitary. Can every unitary on H be written in this way? (b) Let σx, σy, σz be the Pauli matrices. Write down explicitly a unitary matrix U such that UσxU† is diagonal. (Hint: The columns of U must be the eigenvectors of σx. Why?) Do the same for σy and σz. (c) Show that any state |ψ⟩ in H can be written up to a phase in the form |ψ⟩ = e^(iθ)sin(ϕ/2)|1⟩ + e^(iϕ)cos(ϕ/2)|2⟩. (3) Here 0 ≤ θ ≤ 2π. Letting r be the unit vector (sin(ϕ)cos(θ), sin(ϕ)sin(θ), cos(ϕ)), show that (|ψ⟩ = cos(ϕ/2)|1⟩ + sin(ϕ/2)e^(-iθ)|2⟩. (4) (d) Show that the correspondence between unit vectors r and states |ψ⟩ = e^(iθ)sin(ϕ/2)|1⟩ + e^(iϕ)cos(ϕ/2)|2⟩ is one-to-one. What are the states corresponding to r = (1,0,0), (0,1,0), (0,0,1)? (NB: The sphere parameterized by θ and ϕ is in this context called the Bloch sphere.) (e) What is the relationship between the states |ψ⟩ and the eigenvectors of the spin-operator in the x-direction, i.e. σx|ψ⟩ = r⋅σ|r⟩? (f) Find a unitary U such that UU† is equal to the 2 by 2 diagonal matrix diag(1, -1). (g) Show that e^(iσzθ/2) = (phase)(θ), where (phase)(t) means applying to (phase) a rotation by t around the x-axis. Show the corresponding statement for the y and z components.
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Question Text | Texts: Unitary operators, Bloch sphere. In this problem, H = C^2. |1⟩, |2⟩ are the basis vectors.
(1)
Note that |1⟩, |2⟩ are equal to the eigenvectors of H. (a) Show that a linear operator of the form
|1⟩⟨2| + |2⟩⟨1|
(2)
is unitary. Can every unitary on H be written in this way?
(b) Let σx, σy, σz be the Pauli matrices. Write down explicitly a unitary matrix U such that UσxU† is diagonal. (Hint: The columns of U must be the eigenvectors of σx. Why?) Do the same for σy and σz.
(c) Show that any state |ψ⟩ in H can be written up to a phase in the form |ψ⟩ = e^(iθ)sin(ϕ/2)|1⟩ + e^(iϕ)cos(ϕ/2)|2⟩. (3) Here 0 ≤ θ ≤ 2π. Letting r be the unit vector (sin(ϕ)cos(θ), sin(ϕ)sin(θ), cos(ϕ)), show that (|ψ⟩ = cos(ϕ/2)|1⟩ + sin(ϕ/2)e^(-iθ)|2⟩. (4)
(d) Show that the correspondence between unit vectors r and states |ψ⟩ = e^(iθ)sin(ϕ/2)|1⟩ + e^(iϕ)cos(ϕ/2)|2⟩ is one-to-one. What are the states corresponding to r = (1,0,0), (0,1,0), (0,0,1)? (NB: The sphere parameterized by θ and ϕ is in this context called the Bloch sphere.)
(e) What is the relationship between the states |ψ⟩ and the eigenvectors of the spin-operator in the x-direction, i.e. σx|ψ⟩ = r⋅σ|r⟩?
(f) Find a unitary U such that UU† is equal to the 2 by 2 diagonal matrix diag(1, -1).
(g) Show that e^(iσzθ/2) = (phase)(θ), where (phase)(t) means applying to (phase) a rotation by t around the x-axis. Show the corresponding statement for the y and z components. |
Topic | All topics |
Subject | Physics |
Class | Class 12 |