## Section 10.4: Time Evolution

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The time evolution of energy eigenfunctions is governed by the Schrödinger equation,

(−*ħ*^{2}/2*m*) (∂^{2}/∂*x*^{2})Ψ(*x*,*t*) + *V*(*x*)Ψ(*x*,*t*) =* iħ* (∂/∂t) Ψ(*x*,*t*) , (10.5)

and we can write the solution to this equation for the energy eigenfunctions, ψ_{n}(*x*), using Eq. (7.10) as

ψ_{n}(*x*,*t*) = exp(−*iE*_{n}*t*/*ħ*) ψ_{n}(x) . (10.6)

For the infinite square well, we know that ψ_{n}(*x*), given by ψ_{n}(*x*) = (2/*L*)^{1/2} sin(*n*π*x*/*L*) for 0 < *x* < *L* with* n* = 1, 2, 3, …. , is an energy eigenfunction which satisfies the boundary conditions and has an energy eigenvalue *E*_{n}. Therefore the time-dependent energy eigenfunctions (or wave functions) become^{4}

ψ_{n}(*x*,*t*) = exp(−*iE*_{n}*t*/*ħ*) ψ_{n}(x) = (2/*L*)^{1/2} exp[−*in*^{2}π^{2}/*ħ*/(2*mL*^{2}) *t*] sin(*n*π*x*/*L*) (10.7)

using the explicit form of the energy eigenfunction and the energy.

How do we visualize this time-dependent energy eigenfunction? There are several ways. We will demonstrate two. Restart. We can show the real (blue) and/or imaginary (pink) parts of the energy eigenfunction which in our case are just

ψ_{n Re}(*x*,*t*) = cos(*E*_{n}*t*/*ħ*) ψ_{n}(*x*) ,

and

ψ_{n lm}(*x*,*t*) = −sin(*E*_{n}*t*/*ħ*) ψ_{n}(*x*) .

Conversely, when you check the box and "input value and play," we can choose to write the energy eigenfunction as

ψ_{n}(*x*,*t*) = exp(−*iE*_{n}*t*/*ħ*) ψ_{n}(*x*) .

where we have chosen to write all of the complex behavior in an exponential (this is automatically the case here because we have written ψ_{n}(*x*) as a real function). When we do this, −*E*_{n}*t*/*ħ* = θ_{n}(*t*) is an angle in the complex plane and is called the *phase *(or phase angle) of the energy eigenfunction. Note that in the case of energy eigenfunctions of the infinite square well, the phase of the energy eigenfunction does not depend on position. We depict the amplitude of the energy eigenfunction as the magnitude of the distance from the bottom to the top of the energy eigenfunction at a given position and time. We represent the phase as the color of the energy eigenfunction. 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 eigenfunction 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 energy eigenfunction to return to its original phase. In other words, Δ*t* = 1 corresponds to an elapsed time of 2π*ħ*/*E*_{1}.

^{4}This 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 energy eigenfunction, then reform the exponential.

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