10. Schrödinger Equation for Particle in a Time – Independent Potential

The wave function Ψ(x,t) of a particle with mass m, which is moving along the x-direction and is subjected to a time-independent potential V(x), satisfies the Schrödinger equation:

$i\hbar \frac{\partial \Psi }{\partial t}=-\frac{{{\hbar }^{2}}}{2m}\frac{{{\partial }^{2}}\Psi }{\partial {{x}^{2}}}+V\Psi $                                               (10.1)

The Schrödinger equation is a linear and homogeneous differential equation. Consequently, the superposition of each solution is also the solution of the equation. The state at initial time t_o determines its subsequent state at all times, since the Schrödinger equation is a first-order equation with respect to time.

For a particle in a time-independent potential, Schrödinger equation can be solved by the method of separation of variables: the solution that we look for is in the form of a simple product,

$\Psi (x,t)=\psi (x)f(t)$                                                   (10.2)

It can be shown that: $f(t)={{e}^{-\frac{iEt}{\hbar }}}$ with the separation constant E and ψ(x) must satisfy the time-independent Schrödinger equation:

$-\frac{{{\hbar }^{2}}}{2m}\frac{{{d}^{2}}\psi }{d{{x}^{2}}}+V\psi =E\psi $                                                 (10.3)

And we cannot go any further with this equation until the potential V(x) is specified.

Figure 10.1. Separation of variables method.

One of the important characteristics of separable solutions is that the solutions are stationary states. Although the wave-function itself: $\Psi (x,t)=\psi (x){{e}^{{-iEt}/{\hbar }\;}}$ depends on t, the probability density does not depend on t:

$\rho (x,t)=$ |Ψ(x,t)|² = Ψ^*Ψ = ${{\psi }^{*}}{{e}^{{+iEt}/{\hbar }\;}}\psi {{e}^{{-iEt}/{\hbar }\;}}$ = |ψ(x)|²                           (10.4)

And this is valid for the expectation value of any dynamical variables; the general equation for expectation value then reduces to:

 $\left\langle Q(x,p) \right\rangle =\int{{{\psi }^{*}}Q(x,\frac{h}{i}\frac{d}{dx})\psi }dx$                                      (10.5)

Another important property of separable solutions is that: they are states of definite total energy. The total energy (kinetic energy and potential energy) in classical mechanics is called the Hamiltonian:

$H(x,p)=\frac{{{p}^{2}}}{2m}+V(x)$                                           (10.6)

Substituting the momentum p with $\left( \hbar /i \right)\left( \partial /\partial x \right)$ , equation (10.6) becomes an operator:

$\hat{H}=-\frac{{{\hbar }^{2}}}{2m}\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}+V(x)$                                          (10.7)

Therefore, the time-independent Schrödinger equation (10.3) can be written in term of Hamiltonian operator:

$\hat{H}\psi =E\psi $                                                         (10.8)

And the expectation value of the total energy can be calculated:

$\left\langle H \right\rangle =\int{\psi \hat{H}\psi dx}=E\int{{}}$|ψ|² dx = E                                     (10.9)

Furthermore,

${{\hat{H}}^{2}}\psi =\hat{H}(\hat{H}\psi )=\hat{H}(E\psi )=E(\hat{H}\psi )={{E}^{2}}\psi $                               (10.10)

The expectation value of H² is given by:

$\left\langle {{H}^{2}} \right\rangle =\int{{{\psi }^{*}}{{{\hat{H}}}^{2}}\psi dx={{E}^{2}}\int{{}}}$|ψ|² dx = E²                                 (10.11)

The standard deviation in H can be calculated:

$\sigma _{H}^{2}=\left\langle {{H}^{2}} \right\rangle -{{\left\langle H \right\rangle }^{2}}={{E}^{2}}-{{E}^{2}}=0$                                      (10.12)

Therefore, every member of the sample must share the same value, hence we can say that: every measurement of the total energy will certainly return the value of E. This is also one of the important property of a separable solution.

The general solution of time-dependent Schrödinger equation has the property that any linear combination of  separable solutions (ψ₁, ψ₂, ψ₃, ...) is itself a solution. While the time-independent Schrödinger equation yields an infinite collection of separable solutions, each with its associated  value of separation constant, i.e. allowed energy (E₁, E₂, E₃, ...). Then, we can construct a general solution:

$\Psi (x,t)=\sum\limits_{n=1}^{\infty }{{{c}_{n}}{{\psi }_{n}}(x){{e}^{-{i{{E}_{n}}t}/{\hbar }\;}}}$                                        (10.13)

Thus, we simply have to find the right constants (c₁, c₂, c₃, ...) that fit the initial conditions for the problem at hand.