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The time evolution of ψ(x,t) is governed by the Schrödinger equation,
(−ħ2/2m) (∂2/∂x2)ψ(x,t) + V(x)ψ(x,t) = iħ (∂/∂t) ψ(x,t) , (10.12)
and we can write the solution to this equation for the energy eigenstates, ψn(x), as
ψn(x,t) = exp(−iEnt/ħ) ψn(x) . (10.13)
For the infinite square well, we know that ψn(x), given by ψn(x) = (2/L)1/2 sin(nπx/L) for 0 < x < L with n = 1, 2, 3, …. , is an energy eigenstate which satisfies the boundary conditions and has an energy eigenvalue En. Therefore the wave function becomes4
ψn(x,t) = exp(−iEnt/ħ) ψn(x) = (2/L)1/2 exp[−in2π2/ħ/(2mL2) t] sin(nπx/L) (10.14)
using the explicit form of the wave function and the energy.
How do we visualize this time-dependent wave function? There are several ways. We will demonstrate two. Restart. We can show the real (blue) and/or imaginary (pink) parts of the wave function which in our case are just
ψn Re(x,t) = cos(Ent/ħ) ψn(x) , (10.15)
and
ψn Re(x,t) = −sin(Ent/ħ) ψn(x) . (10.16)
Conversely, when you check the box and "input value and play," we can choose to write the wave function as
ψn(x,t) = exp(−iEnt/ħ) ψn(x) . (10.17)
where we have chosen to write all of the complex behavior in an exponential (this is automatically the case here because we have written ψn(x) as a real function). When we do this, −Ent/ħ = θn(t) is an angle in the complex plane and is called the phase (or phase angle) of the wave function. Note that in the case of energy eigenstates of the infinite square well, the phase of the wave function does not depend on position. We depict the amplitude of the wave function as the magnitude of the distance from the bottom to the top of the wave function at a given position and time. We represent the phase as the color of the wave function. The color strip above the animation shows the map between phase angle and color. Since quantum-mechanical time evolution involves a minus sign in the exponential, the phase evolves in time counterclockwise in the complex plane.
Explore the time dependence of the energy eigenstate of a particle in an infinite square well by changing state and representation. In the animation the time is given in terms of the time it takes the ground-state wave function to return to its original phase. In other words, Δt = 1 corresponds to an elapsed time of 2πħ/E1.
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4This result can be derived by a Taylor series expansion of the exponential in the time-evolution operator and operate the successive powers of the Hamiltonian on the wave function, then reform the exponential.