How does the phase of an eigenstate evolve over time according to Schrodinger's equation?
According to Schrödinger's equation, the phase of an eigenstate evolves over time in a deterministic manner. The equation, named after Austrian physicist Erwin Schrödinger, is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It is a partial differential equation that relates the time derivative of the wave function to its spatial derivatives and the potential energy of the system.
To understand how the phase of an eigenstate evolves, let's first clarify what an eigenstate is. In quantum mechanics, an eigenstate is a state of a system that satisfies a specific eigenvalue equation. The eigenvalue equation is obtained by applying an observable operator to the wave function and obtaining a constant multiple of the original wave function. The eigenvalue represents the value that will be obtained when measuring the corresponding observable.
In the context of Schrödinger's equation, the time evolution of a quantum system is governed by the Hamiltonian operator, which represents the total energy of the system. When the Hamiltonian operator is applied to an eigenstate, it yields the corresponding eigenvalue multiplied by the eigenstate itself. Mathematically, this can be expressed as:
H |ψ⟩ = E |ψ⟩
where H is the Hamiltonian operator, |ψ⟩ is the eigenstate, E is the eigenvalue, and the symbol "⟩" represents a ket vector in Dirac notation.
The time evolution of the eigenstate |ψ⟩ is given by the time-dependent Schrödinger equation:
iħ ∂/∂t |ψ(t)⟩ = H |ψ(t)⟩
where i is the imaginary unit, ħ is the reduced Planck's constant, and ∂/∂t represents the partial derivative with respect to time.
To solve this equation, we can express the time-dependent wave function |ψ(t)⟩ as a linear combination of the eigenstates of the Hamiltonian operator:
|ψ(t)⟩ = Σ C_n |ψ_n⟩
where C_n are complex coefficients and |ψ_n⟩ are the eigenstates of the Hamiltonian operator.
Substituting this expression into the time-dependent Schrödinger equation, we obtain:
iħ Σ ∂/∂t (C_n |ψ_n⟩) = Σ C_n (H |ψ_n⟩)
Expanding the derivatives and using the fact that the eigenstates are orthogonal, we get:
iħ Σ (C_n ∂/∂t |ψ_n⟩) = Σ C_n (H |ψ_n⟩)
Since the eigenstates are orthogonal, the terms on the right-hand side can be simplified to:
iħ Σ (C_n ∂/∂t |ψ_n⟩) = Σ C_n (E_n |ψ_n⟩)
where E_n are the eigenvalues of the Hamiltonian operator.
Now, we can separate the terms corresponding to each eigenstate:
iħ (C_1 ∂/∂t |ψ_1⟩ + C_2 ∂/∂t |ψ_2⟩ + … ) = C_1 (E_1 |ψ_1⟩) + C_2 (E_2 |ψ_2⟩ + … )
Since the eigenstates are orthogonal, we can multiply both sides of the equation by the complex conjugate of an eigenstate and integrate over all space:
iħ ∫ (C_1* ∂/∂t |ψ_1⟩ + C_2* ∂/∂t |ψ_2⟩ + … ) dV = ∫ (C_1* E_1 |ψ_1⟩ + C_2* E_2 |ψ_2⟩ + … ) dV
Using the orthonormality of the eigenstates, we find:
iħ (C_1* ∂/∂t ∫ |ψ_1⟩ dV + C_2* ∂/∂t ∫ |ψ_2⟩ dV + … ) = C_1* E_1 ∫ |ψ_1⟩ dV + C_2* E_2 ∫ |ψ_2⟩ dV + …
The integrals on the left-hand side are just the inner products of the eigenstates with themselves, which are equal to 1:
iħ (C_1* ∂/∂t + C_2* ∂/∂t + … ) = C_1* E_1 + C_2* E_2 + …
Simplifying further, we obtain:
iħ (dC_1/dt + dC_2/dt + … ) = C_1* E_1 + C_2* E_2 + …
This is a set of coupled first-order ordinary differential equations, known as the time-dependent Schrödinger equation in the eigenstate representation. Solving these equations allows us to determine how the complex coefficients C_n evolve over time, and hence the time evolution of the eigenstate |ψ⟩.
The phase of an eigenstate evolves over time according to Schrödinger's equation by determining the time-dependent complex coefficients associated with each eigenstate. The specific evolution of the phase depends on the eigenvalues and eigenstates of the Hamiltonian operator, which describe the energy and spatial distribution of the quantum system.
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