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The Time Dependent Schrödinger Equation
and the Nature of its Solution

The time dependent Schrödinger equation for a system is

ih(∂ψ/∂t) = H^ψ

where H^ is the operator derived from the Hamiltonian function for the system. The symbol h is Planck's constant divided by 2π, t is time and i denotes the square root of negative one. The variable ψ is called the wave function and its nature is in dispute.

The Correspondence Principle

Neils Bohr nearly a century ago observed that classical analysis for many areas of physics had been empirically verified. Therefore, for any quantum mechanical analysis its appropriate extension to the realm of classical analysis should agree with the classical analysis. In atomic physics the extension is in terms of scale and/or the level of energy. In radiation physics the extension is the limit as h, Planck's constant goes to zero. In statistical mechanics the limit as the number of molecules increases without bound should agree with thermodynamics.

Thus in order for a quantum mechanical analysis to be valid it must obey the Correspondence Principle. For an example of an analysis involving the time dependent Schrödinger equation consider a particle traveling freely in 3D space. The solution to the time dependent Schrödinger equation has the particle spread uniformly over an infinite plane perpendicular to its direction of motion. Not only is this an unacceptable probability distribution but also it does not satisfy the Correspondence Principle. Nothing in the solution gives an asymptotic approach to the concentration of the particle to a limited volume of space in the classical analysis.

A Particle in a Central Potential Field

Let V(r) be the potential energy of a particle as a function of its distance r from the center of the potential field. The Hamiltonian function for this system is

H = p²/(2m) + V(r)

where m is the mass of the particle and p is its momentum. In polar coordinates (r, θ)

p² = m(dr/dt)² + m(r(dθ/dt))²

According to the rules formulated by Schrödinger p² is replaced in the Hamiltonian function by −h²∇² to obtain the Hamiltonian operator for the system. The time-dependent Schrödinger equation for the system is

ih(∂ψ/∂t) = −h²∇²ψ + V(r)ψ

At this point in quantum analysis it is customary to apply the separation of variables technique. This technique is not innoculous and could preclude finding the physically relevant solutions to the equation. It is worth applying if for no reason other than to gain some insights to the physical system. Furthermore this technique is valid for some simple, symmetric cases.

The separation of variables technique assumes that ψ(r, θ, t) = S(r, θ)T(t). Thus the Schrödinger equation becomes

ihS(r, θ)T'(t) = −h²(∇²S(r, θ))T(t) + V(r)S(r, θ)T(t)
and, upon division by ST
ihT'(t)/T = −h²(∇²S(r, θ))/S + V(r)

The LHS of the above equation is a function only of t, the RHS a function only of r and θ. Therefore the common value of the LHS and RHS must be a constant, say E.

This means

ihT'(t)/T = E
and thus
T'(t)/T = −iE/h

Therefore

T(t) = T(0)exp(−i(E/h)t)

This is an oscillatory solution whose magnitude is constant.

It is also true that

h²(∇²S(r, θ))/S + V(r) = E
or, equivalently
h²(∇²S(r, θ)) + V(r)S = ES

This is equivalent to the time-indepent Schrödinger equation. It is known from previous studies that the solution to this equation implies a probability density function that is inversely proportional to velocity and thus the time spent in a state.

Generalization

The above procedure could just as well have been in the form of a system with generalized coordinates {q1, …, qn}

The time-dependent Schrödinger equation is then

ih(∂ψ/∂t) = H^ψ

If ψ is assumed to be of the form Ψ(q1, …, qn)T(t) then

hΨT'(t) = T(H^Ψ)
and thus
ihT'(t)/T = (H^Ψ)/Ψ

Again the LHS is a function only of t and the RHS only of the generalized coordinates. The common value of the two sides must be a constant, say E.

Thus, as before,

T(t) = T(0)exp(−i(E/h)t)

and Ψ satisfies the time-independent Schrödinger equation

H^Ψ = EΨ

The Special Case of ∂ψ/∂t Being Equal to Zero

This corresponds to a particle with no motion and thus the probability density function of its location is a Dirac delta function.

Conclusion

The valid solutions to the time dependent Schrödinger equation correspond to probability density functions which are the proportions of the time spent in the allowable states of the system.

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