So far, we have made a lot of progress concerning the properties and interpretation of the wave function, but as yet we have had very little to say about how the wavefunction may be derived in a general situation, that is to say, we do not have on hand a “wave equation” for a wave function. There is no true derivation of this equation, but its form can be motivated by physical and mathematical arguments at a wide variety of levels of sophistication. Here, we will offer a simple derivation based on what we have learned so far about the wave function.
The Schrodinger equation has two forms’, one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. In general, the wave function behaves like a, wave, and so the equation is, often referred to as time dependent Schrodinger wave equation. The other is the equation in which the time dependence has been removed and hence is known as the time independent Schrodinger equation and is found to describe, amongst other things, what the allowed energies are of the particle.
These are not two separate, independent equations the time independent equation can be derived readily from the time dependent equation.
9.9.1 Derivation of the Schrodinger Wave Equation,
(a) The time Dependent Schrodinger Wave Equation; Let us consider a microparticle. Let Ψ be the wave function associated with, the motion of this microparticle Ψ function represents the wave field of the particle. It is similar to E and B used to describe the electromagnetic waves and to transverse displacement for waves on a string:
For one-dimensional case, the Classical wave equation has the following form.
(b) The Time Independent Schrodinger Equation: In many cases the potential energy
V of a particle does not depend on time, it varies only with position of the particle and field is said to be stationary. In such cases Schrodinger time independent wave equation is used.
In this case the wave function is written ,as product of Ψ(x) and Ψ(t)
Ψ(x,t) = Ψ(x) Ψ(t) … (1)
Substituting this in Schrodinger’s time dependent wave equation
The k vector describes the wave properties of the electrons.
As E k2, the graph between E and k is a parabola as visualised in Figure 9.7.
FIGURE 9.7 Parabolic relationship between E and k in case of free electron
The momentum is well defined in this case. Therefore according to uncertainty
principle it is difficult to assign a position to the electron, i.e., the electron position is in determinate.
ii. Infinite Potential Well: A Confined Particle in One Dimensional Box: A particle is trapped in one dimensional box according to the following potential distribution box according to the following potential distribution
are allowed energy states. It is shown in Figure 9.9
Energy of particle in one dimensional box can take only discrete values i.e., it is quantized. The value of energy E1 for n = 1 is called zero point energy, which signifies that there must be some movement of particles (atoms, molecules etc.) at absolute zero temperature.
- Equation (5) shows that energy is a function of quantum number and width of the well.
- The particle (like electron) trapped in the potential well (box) cannot take zero energy. Because if, energy is zero, momentum also would be zero and the uncertainty principle requires that its wavelength λ be , If its wavelength is infinity, then it cannot confired to a box. Therefore the electron must possess a certain minimum amount of kinetic energy.
iii. Potential Step: Reflection and Tunnelling: Quantum Leak: Let us consider a particle is moving along positive x-axis towards a finite potential step.
We see that this probability is finite (as E<V0). Therefore the probability of finding the particle in the forbidden region of positive x can be done by integrating the wave function over the x values from 0 to , which will be finite and proportional to . This is an example of tunneling effect, a particle can be sometimes found where its energy is negative (E<V0) and therefore, classically forbidden. Actually, wave penetrating a small distance from the boundary into region II is continually reflected till all the incident energy is turned back into region I. Due to such continual reflection, the amplitude of the wave penetrating into region II falls off exponentially. Classically, a particle of energy E < V0 can never penetrate into region II, since its K.E. would be negative, but in quantum physical world there is a finite probability of the particle wave penetrating a short distance into the classically forbidden region II and decaying exponentially.
Now for the case of E > V0 according to equation (6), |C| > |A| since k1 > k2. So the amplitude of the transmitted wave is greater than that of incident wave. Also, as the kinetic energy of the particle is greater in region I, therefore, the de Broglie wavelength is shorter in region I with respect to region II as explained in Figure 9.12. On the other hand, for the case of E < V0 the transmitted wave decays exponentially within a very short distance in region II as explained earlier and shown in Figure 9.12.
FIGURE 9.12 Wave functions for a stop potential