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The Hamiltonian of a system with two states is given by the following expression: ħwoox H where ôx = And the eigen vectors for this Hamiltonian are: |-) = [d |+) = H Whose eigen values are E₁ ħwo and E₂ -ħwo respectively. We initialise the system in the state. |(t = 0)) = cos (a)|+) + sin(a)|-)

The Hamiltonian of a system with two states is given by the following expression: ħwoox H where ôx = And the eigen vectors for this Hamiltonian are: |-) = [d |+) = H Whose eigen values are E₁ ħwo and E₂ -ħwo respectively. We initialise the system in the state. |(t = 0)) = cos (a)|+) + sin(a)|-)

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The Hamiltonian of a system with two states is given by the following expression: H = ħwoox where x = [₁¹] And the eigen vectors for this Hamiltonian are: |+) 調 1-) = √2/24 Whose eigen values are E₁ = = = ħwo and E₂ We initialise the system in the state. (t = 0)) = cos (a)|+) + sin(a)|-) -ħwo respectively. a. Show the initial state |(t = 0)) is correctly normalised. b. For this problem write the full time-dependent wavefunction. C. If the system starts in the state |(t = 0)) = (1+) + |−))/√√2 (i.e., a = π/4), what is the probability it will stay in this state as a function of time?
Transcribed Image Text:The Hamiltonian of a system with two states is given by the following expression: H = ħwoox where x = [₁¹] And the eigen vectors for this Hamiltonian are: |+) 調 1-) = √2/24 Whose eigen values are E₁ = = = ħwo and E₂ We initialise the system in the state. (t = 0)) = cos (a)|+) + sin(a)|-) -ħwo respectively. a. Show the initial state |(t = 0)) is correctly normalised. b. For this problem write the full time-dependent wavefunction. C. If the system starts in the state |(t = 0)) = (1+) + |−))/√√2 (i.e., a = π/4), what is the probability it will stay in this state as a function of time?
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