LECTURE 10. WAVE MECHANICS AND SCHROEDINGER’S EQUATION
Topics
- Concepts of a wave function - Introduction to Schroedinger’s equation for a free particle - Application of Schroedinger’s equation to a particle constrained in a box - Time independent Schroedinger’s equation - Ehrenfest Theorems and applications - Basic postulates - Eigenvalues and eigenfunctions - Degeneracy
OBJECTIVES
At the end of the lecture you should be able to
1) Explain the meaning of the wave function Ψ, and state its properties.
2) State and show steps leading to the time- dependent Schrödinger equation of a free particle
3) Write down the time-dependent Schrödinger equation for a restricted particle.
4) Verify the correct wave functions for free and confined particles.
5) Solve simple problem involving the wavelength Ψ.
6) Show that the energy of a particle confined to a potential well is quantized, hence draw the appropriate energy level diagram.
7) Define eigenvalues, eigenstates and eigenfunctions
8) Explain the concept of degeneracy.
9) For trapped particles, state the features that are predicted by quantum theory but not by classical theory.
10.1 Concepts of a wave function
We have already discussed in the previous lecture the fact that a moving particle can be considered as a wave, with specific wavelength, which is inversely proportional to the magnitude of the momentum p. In Mechanics or Waves and Optics courses, you have learnt that in a medium the wave disturbance in the x-direction can be represented by the transverse displacement (y).
Recall the wave equation
2 y 2 y c 2 (10.1) 2 2 t x where c is a constant with the dimensions of velocity.
Equation (10.1) is a differential equation. Similarly in Wave Mechanics ( a semi-quantum theory constructed to solve problems involving electrons and other sub-atomic particles, for which classical laws were found to be deficient) we look for an equation that can be used for the wave s representing particles. For this purpose we introduce a new wave function (), which at every point in space and time can be written as (x,y,z,t). This wave function has the following characteristics:
1) , in general, is not real but a complex function, that it is, it has a real part and an imaginary part. (Recall what real and complex numbers are). This means that the wave function should be interpreted in terms of probabilities instead of certainties. Therefore the square of is interpreted as the probability (per unit volume) that the particle will be found at that point. The probability that a particle is somewhere (in volume element dV) is unity so that
2 dV 1 (10.2) 2) is not a measure of any simple physical entity such as displacement. 3) is continuous in space. That is, the wave function should not have breaks in it for the solution to be valid.
We note that in Wave mechanics, physical information about the particles is contained in complex wave functions from which probability predictions can be got.
10.2 The Schrödinger Equation
The Schrödinger theory of quantum mechanics specifies the laws of wave motion that the particles that comprise microscopic systems obey. (In Newtonian Mechanics the Newton’s second law, and in electromagnetic theory the Maxwell’s equations serve this role). This is achieved by stating, for each system, the equation that controls the behaviour of the wave function , and also by giving the relation between the behaviour of the wave function and the behaviour of the particle. Schrödinger’s theory is therefore an extension of the de Broglie’s postulate. It also bears close resemblance of Newton’s theory of motion of particles in macroscopic systems.
It is worth noting that the de Broglie’s postulate only says that the motion of a microscopic particle is governed by the propagation of an associated wave. It never says how the wave propagates. It was Erwin Schrödinger’s (1926) who developed an equation that tells us the behaviour of any wave function of interest.
We shall assume a free particle, i.e. one that is not acted on by an external force, and further, that the motion is one-dimensional.
Now consider a particle of momentum P and energy E. Its wavelength is given by h and p frequency by E . We can define h 2 P Wave number k (10.3) And E Angular frequency 2v (10.4) 2 For a free particle E P , thus 2m 2mE k (10.5)
Now if the particle travels in the x-direction, the wave will also travel in the same direction. The harmonic wave (correct wave function) meeting this condition is
x,t e it kx for motion in the positive x-direction (10.6) and x,t e it kx for motion in the negative x-direction (10.7)
Note that in arriving at the correct harmonic wave function we have invoked the superposition principle of quantum mechanics: that is, if 1 and 2 are possible wave functions, then so is
1 2 .
We now want to find out the wave equation obeyed by these wave functions. We rewrite Eq. (10.6) by expressing and k in terms of P, that is,
Et x i 2mE eit kx e (10.8)
We differentiate Eqn. (10.8) once with respect to time (t) and twice with respect to x to obtain
E i (10.9) t and
2 2mE 2 2 (10.10) x
Comparing equations (10.9) and (10.10) we notice that
(10.11) 2 2 2 x,t i 2 x,t 2m x t
This is the wave equation for a free particle; and it is called The-time dependent Schrödinger Equation in one dimension.
If the particle is restricted the presence of a force is represented by some given potential energy function (V(x,t)), equation (10.11) now takes the form
2 2 V x,t (10.12) i t 2m t 2
(We have used the fact that (i 2 1 )
Exercise Generalize equation (10.12) into the three dimensions form.
The general Schrödinger Equation is the basic equation of quantum mechanics. It serves as the equation of motion for the wave function and it allows us to calculate the wave function at any later time given the initial conditions at some initial time.
Of the solutions to the Schrödinger equation, the standing-wave solutions are of the most importance. Wave mechanics explains the quantization of energy in terms of standing waves. Just as the possible standing waves in an air column or in a stretched string (see Fig. 10.1) must have an integral number of wavelengths or half-wavelengths with the air column or stretched string so does the discrete set of frequencies and energies of a quantum-mechanical particle bound within a system arise.
Figure 10.1: The first four modes of vibration of a stretched string fixed at both ends.
10.3 Particle in a Box
One simplest way of restricting a quantum-mechanical system is to restrict it to a one- dimensional ‘box’. The walls of the box are assumed to be perfectly elastic, and rigid, and the particle moves freely between these walls. V(x)
0 L x
Figure 10.2 Potential energy for s particle in a one-dimensional box. The potential energy is infinite at x – 0 and x = L.
If the particle has energy E and momentum P 2mE, then the harmonic waves traveling to the right and the left along the x-axis are, respectively,
Et x i 2mE e (10.13) and Et x i 2mE e (10.14)
Employing the superposition principle (from equations (10.13) and (10.14) we obtain
Et x Et x i 2mE i 2mE x,t Ae Be (10.15) where for the sake of generality we have multiplied Eqns. (10.13) and (10.14) by constant A and B respectively. A standing wave will result provided A B.
At x =0 and at x = L, ψ must be zero at all times. Thus
0,t 0 at x 0 (10.16) and L,t 0 at x L (10.17)
Equations (10.16) and (10.17) are the boundary conditions. These boundary conditions are the same as those for a vibrating string fixed at both ends.
From equations (10.16) and (10.17) we obtain from equation (10.15)
AeiEt / BeiEt / 0 at x = 0 (10.18) and Et L L i 2mE i 2mE Ae Be 0 at x = L (10.19)
From equation (10.18) we obtain B = -A, and therefore equation (10.19) reduces to
L L i 2mE i 2mE iEt / Ae e e 0 (10.20)
This equation is equal to
L 2iA sin 2mE 0 (10.21) from the fact that
ei ei sin 2i
For the wave function to vanish (become zero) at the boundaries (see Fig. 10.2) we demand that the argument of the sine function be an integer multiple of π,
Remember Sin = 0 if = 0, , 2, 3,4 etc., where = 180
L 2mE n (10.22)
or
2 2 2 n n = 1,2,3 (10.23) En 2 2mL
Equation (10.23) shows that the energies of the particles in a box are quantized. The lowest 2 2 2 energy is obtained by setting n = 1, and is the energy of the ground state E1 /2mL Shown in Figure 10.3 are the energy levels for a particle in a box as predicted by equation (10.23). The quantized energies (E1, E2, E3, etc) of the systems are called energy eigenvalues, and the states that correspond to the definite values of quantized energy are called energy eigenstates. Corresponding to each eigenvalue is an eigenfunction ( 1 , 2 , 3 , etc).
(a) (b)
Fig. 10.3 Energy-level diagram for a particle in (a) 1-D box (b) 3-D box (see section 10.7)
The notion of eigenvalues affords a way in which the results of measurements can be associated with quantum mechanical observables. Such observables include position, momentum and energy.
Since we have shown that B = -A, Eqn (10.15) can be rewritten as
Et x Et x 1 2mE i 2mE x,t Ae Ae Et x Et x i i 2mE i i 2mE Ae e Ae e Et x Et i i 2mE i 2mE Ae e e
n2 2 2t n2 2 2m.n2 2 2 i ix ix 2 2 2 2 Ae 2mL e L e 2mL in2 2 t n n 2 ix ix Ae 2mL e L e L
2 2 in t 2 n (x,t) 2i Ae 2mL sin (10.24) L
2 in t Notice that eqn (10.24) consists of a time-dependent part e 2mL2 and space (x)-dependent part. Let us how consider only the spatial part x, i.e., that part of the general wave function that nx contains (x) only. This we can write as n x Asin ,n 1,2,3 . The constant A L can be determined by the normalization condition and found to be
2 A (10.25) L So that
2 nx n x sin (10.26) L L
2 The square of the function n x , i.e., n x is the probability density Pn x .
2 2 2 nx Pn x n x A sin ; n,1,2,3 10.27 L
For a particular (x), Pn(x) simply gives the probability of an electron or particle being located there. The spatial wave function (x) and the probability density Pn(x) for some cases are shown in Fig .10.4 and 10.5. Notice that the number of crests equals n, in each case
Fig. 10.4 Space dependence of Fig. 10.5 The probability density Pn(x) calculated
n(x) from Eqn. (10.27)
10.3.1 Predictions of quantum theory regarding a trapped particle
1. Energy is quantized 2 2 h 2 E n 2 n 2 n 2mL2 8mL2 2. If the trapped particle is an electron, the electron cannot be at rest in the potential h 2 well. The lowest energy is E is called the zero-point energy of the infinite 1 8mL2 potential well. This implies that even at the absolute zero temperature the particle is still in motion and has energy.
3. The electron spends more time in certain parts of the well than in others but for states of higher quantum number, the electron probability density becomes more uniform and quantum prediction merges with the prediction of classical physics.
4. The electron can escape from its trap if the trap has finite potential. This phenomenon is known as tunneling.
10.4 Time-independent Schrödinger’s equation
There is another form of the Schrödinger’s equation known as the Schrödinger’s time- independent equation. For a free particle in one-dimension (x-axis) it is of the form
2 2mE 2 2 0 (10.28) x
For an unrestrained quantum-mechanical particle it has the form
2 2mE 2 2 E V 0 (10.29) x
10.5 Expectation Values
As we have already noted that in the quantum theory the wave function ψ can only make a probabilistic prediction. Since we cannot make definite predictions as to which position or momentum will be measured; we can only give probabilities. If we repeat the measurement of position or of momentum several times and calculate the mean value, the latter may now be defined precisely. This mean value is therefore called the expectation value; and it is defined thus: Expectation value = Sum over individual values measured, times the probability that the value would be found.
The expectation (average) value of position ( , and energy 2 x x x dx (10.30) x,t P i x,t dx (10.31) x where ψ* is the complex conjugate of ψ. * x,t E i x,t dx (10.32) x 10.6 Ehrenfests’ Theorem Ehrenfest (1927) showed that the motion of a wave packet obeys Newton’s laws. That is, he showed that if are the expectation (average) values of the position and momentum of the wave packet then d m x P (10.33) dt x and if V is the potential energy of the particle d V P (10.34) dt x x For a sharply defined wave packet 10.7 Degeneracy In 3-dimensions, the time-independent (steady-state form of) Schrödinger’s equation is 2 2 2 2m 2 2 2 2 E 0 (10.35) x y z The simplest 3-D box is an extension of the 1-D box discussed earlier, i.e., a cube of length L with infinitely rigid wall. Such a box corresponds to a potential energy V x, y, z whose value is V = 0 inside the box and V = outside it. Thus inside the box the Schrödinger’s equation becomes 2 2 2 2m 2 2 2 2 E 0 (10.36) x y z We solve equation (10.32) to obtain ψ subject to the conditions ψ = 0 at the sides of the box. This equation is solved by first separating it into three independent equations which involves only one coordinate. This is achieved by assuming that the wave function x, y, z is the product of three different functions: x x, which depends on x only, y x, which depends on y only, and zz which depends on z only. That is, x,y, z x x y y z z (10.37) This assumption means that the variation in ψ in one direction is independent of variations in the other directions. With this assumption and after some mathematical manipulations (taking partial derivatives of x y z , and substituting the results in Eq. (10.36), dividing through x y z , and rearranging) gives 2 2 1 d x 1 d z 2mE x 2 2 2 (10.38) dx y dz But since the variation of ψ is independent in each direction, therefore the terms on the left hand side can be separated and set equal to independent constants: 2 1 d x 2 2 k x (10.39) x dx 2 1 d y 2 2 k y (10.40) y dy 2 1 d z 2 2 k z (10.41) z dz such that 2 2 2 2mE k x k y k z 2 (10.42) Imposing the boundary conditions that ψ = 0 at the wall of the box, we realize that the only acceptable solutions to equations (10.39) to (10.41) which are all differential equations is the sine function. The complete wave function therefore becomes x y z Asin k x x sin k y ysin k z z (10.43) where A is a normalization constant whose value is obtained from the fact that 2 dV 1 Now for ψ to be zero at x, y, z = L (for a cube of size L), we set k x L nx nx 1,2,3..... k y L n y n y 1, 2,3... k z L n2 nz 1,2,3.... Hence the wave function in a cubical box becomes n x n y n z Asin x sin y sin z (10.44) L L L and the possible energies of the particle are 2 2 E n 2 n 2 n 2 (10.45) x y z 2mL2 We now notice that three quantum numbers nx ,n y and nz are now required to specify each state of a particle, unlike in the case of the 1-D where one quantum number is required. Consider these scenarios for which nx 2 nx 1 nx 2 n y 2 n y 2 n y 1 nz 1 nz 2 nz 2 2 2 2 In all these three quantum states the confined particle has energy of 9 / 2mL . An energy level of this kind is said to be degenerate. In this example we have the three-fold degeneracy. As another example where different sets of the quantum numbers nx ,n y and nz give rise to the same energy E, consider the case of (1,2,3), (2,1,3), (3,1,2), (1,3,2), (2,3,1) and (3, 2,1). These 2 2 2 are six quantum stationary states all with same energy 14 / 2mL .This is an example of the six-fold degeneracy. See fig. 10.3b for the energy level diagram. SUMMARY 1. Interpretation of wavefunction: Probability for an electron to be found between x and x+dx (x,t) 2 dx 2. Energies of stationary states of particle in a 1-D box: 2 2 h 2 2 2 En n 2 n 2 2mL 8mL 3. The eigenfunction for the stationary states: 2 2 in t 2 2mL2 n (x,t) e sin L L 4. Time dependent Schrödinger Equation 2 2 2 x,t i 2 x,t 2m x t 5. Time independent Schrödinger Equation for unrestrained particle 2 2mE 2 2 E V 0 x 6. Energy of stationary states of a particle in 3-D box 2 2 E n 2 n 2 n 2 x y z 2mL2 PROBLEMS 1. Why is the wave nature of matter not more apparent in our daily experiences? 2. The wave nature of electrons has been utilized to construct an ‘electron microscope’ using short-wavelength electrons to provide high resolution. (a) How might the electron beam be focused? Is it possible to construct a proton or neutron microscope? 3. Is an electron a particle? Is it a wave? Explain your answer citing relevant experimental evidence. [You may have to read a number of textbooks to answer this question adequately]. 4. The quantity x , the amplitude of a matter wave is called a wave function. What is the relationship between this quantity and the particles that form the matter wave? 5. A 1 x 10-9 kg speck of dust is confined between two rigid walls that are 1 x 10-4m apart. If the speck takes 100S to cross this gap. Determine the quantum number applicable this situation. Can the quantum nature of this speck be revealed? [Ans: 3 x 1014]. Explain your answer. 6. Calculate the energies, in eV, of the three lowest states of an electron confined by electrical forces to an infinitely deep potential well whose length is 1 x 10-10m. (Note this length is approximately one atomic diameter). What is the energy of the state with n = 15? [Ans: E1 = 37.7eV, E2 = 150.8 eV, E3 = 339.3 eV, E15 = 8480 eV] 7. Schrödinger’s time independent equation for a free particle confined to the x-axis is 2 d 2mE 2 2 0 dx Show that x i 2mE i 2mE Ae Be x Where A and B are constants is a solution of this equation. 8. A particle in a box is in the first excited state, described by the equation 2 in2 t /2mL2 nx x,t sin with n = 2. What is the L L probability of finding the particle in the infinitesimal interval dx 0.001L near x 1 L? In a similar interval near x 1 L ? 4 3 [Ans. 2 x 10-3, 1.5 x 10-3] SOME REFERENCES 1. Beiser, A., Perspectives of Modern Physics, International Student Edition, McGraw-Hill Kogakusha, Ltd, Tokyo, 1969. 2. Clark, H., A First Course in Quantum Mechanics, English Language Book Society and Van Nostrand Reinhold Company Ltd, London, 1982. 3. Haken, K and H.C. Wolf, The Physics of Atoms and Quanta: An Introduction to Experiments and Theory (5th edn), Springer, Berlin, 1996. 4. Halliday, D, R. Resnick and K.S. Krane, Physics Vol II (5th edn), John Wiley and Sons, Singapore, 1992. 5. Ohanian H.C., Modern Physics (2nd edn), Prentice-Hall International Inc., New Jersey, 1995.