How can the time evolution of the qubit state be computed using the eigenvalues of the Hamiltonian for Larmor precession?
The time evolution of a qubit state can be computed using the eigenvalues of the Hamiltonian for Larmor precession. To understand this, let's first discuss the concept of a qubit and the Hamiltonian.
In quantum information, a qubit is the fundamental unit of information. It is a two-level quantum system that can be represented as a superposition of two basis states, usually denoted as |0⟩ and |1⟩. These basis states can correspond to any two orthogonal quantum states, such as the spin-up and spin-down states of a particle.
The Hamiltonian, in the context of quantum mechanics, represents the energy of a system and governs its time evolution. In the case of Larmor precession, the Hamiltonian describes the precession of a qubit's spin around an external magnetic field. It is given by:
H = ωσz/2
Here, H is the Hamiltonian, ω is the Larmor frequency (proportional to the strength of the external magnetic field), and σz is the Pauli matrix that represents the spin along the z-axis.
To compute the time evolution of the qubit state, we need to solve the time-dependent Schrödinger equation:
iħ d/dt |Ψ(t)⟩ = H |Ψ(t)⟩
where ħ is the reduced Planck's constant and |Ψ(t)⟩ is the state of the qubit at time t.
To solve this equation, we can use the eigenvalues and eigenvectors of the Hamiltonian. The eigenvectors of the Hamiltonian represent the stationary states of the qubit, while the corresponding eigenvalues determine the energy associated with each state.
Let's denote the eigenvectors of the Hamiltonian as |+⟩ and |-⟩, corresponding to the eigenvalues E+ and E-, respectively. These eigenvectors are given by:
|+⟩ = cos(θ/2) |0⟩ + e^(iϕ) sin(θ/2) |1⟩
|-⟩ = -e^(-iϕ) sin(θ/2) |0⟩ + cos(θ/2) |1⟩
Here, θ and ϕ are parameters that depend on the initial state of the qubit and the Larmor frequency.
The time evolution of the qubit state can be expressed as a linear combination of the eigenvectors:
|Ψ(t)⟩ = c+(t) |+⟩ + c-(t) |-⟩
where c+(t) and c-(t) are the probability amplitudes associated with the eigenvectors |+⟩ and |-⟩, respectively.
By substituting this expression into the time-dependent Schrödinger equation, we obtain a system of coupled differential equations for the probability amplitudes:
iħ dc+/dt = E+ c+(t)
iħ dc-/dt = E- c-(t)
The solutions to these equations are given by:
c+(t) = c+(0) e^(-iE+t/ħ)
c-(t) = c-(0) e^(-iE-t/ħ)
where c+(0) and c-(0) are the initial probability amplitudes.
From these solutions, we can compute the time evolution of the qubit state by plugging them back into the expression for |Ψ(t)⟩. This allows us to determine the probability of finding the qubit in the basis states |0⟩ and |1⟩ at any given time.
The time evolution of a qubit state can be computed using the eigenvalues of the Hamiltonian for Larmor precession. By solving the time-dependent Schrödinger equation and expressing the qubit state as a linear combination of the eigenvectors of the Hamiltonian, we can determine the probability amplitudes and compute the time evolution of the qubit state.
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