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29 lines
1.4 KiB
Plaintext
29 lines
1.4 KiB
Plaintext
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A spin-1/2 particle with a magnetic moment is known to be in the state |Ψ(t=0)〉 = |+〉.
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a) If the observable S𝓍 is measured at t=0, the possible results are ħ/2 and -ħ/2 with an equal probability of measuring either.
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b) The system evolves in a uniform magnetic field 𝐁 = B₀ŷ. What is the state of the system at t=T?
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Since the magnetic field is oriented along the y axis, the energy eigenstates will be associated with that direction. The eigenstates are |±𝓎〉.
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The Hamiltonian for this system is
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H = - ω₀ S𝓎, with ω₀ = g q/2mₑ B₀ ≈ e/mₑ B₀.
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The eigenvalue equations in the energy basis are therefore
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Ĥ|E±〉 = ω₀ ±ħ/2 |E±〉 = ω₀ ±ħ/2 |±〉 = E± |±〉
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The initial state is prepared to |+〉, which means, in the y basis,
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|Ψ(t=0)〉 = 1/√2 |+𝓎〉 - 1/√2 |-𝓎〉
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This Hamiltonian is time independent, so the time evolution is given by multiplying each eigenstate with the time-pdependent phase factor, with ι the imaginary unit.
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|Ψ(t)〉 = exp(-ι E₊ t/ħ)/√2 |+𝓎〉 - exp(-ι E₋ t/ħ)/√2 |-𝓎〉
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If the time-evolution of the general state is known in the energy basis (aligned with the spin y basis), the time-evolved state at some time t in the z basis can be predicted by projecting the z basis onto the state to determine the coefficients in the z basis. In mathematical terms,
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〈+|Ψ(t)〉|+〉 =
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〈-|Ψ(t)〉|-〉 = (
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