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Home , Matter wave, Mechanics, Motion, Quantum mechanics

LECTURE 10. AND SCHROEDINGER’S

Topics

- Concepts of a - Introduction to Schroedinger’s equation for a free - Application of Schroedinger’s equation to a particle constrained in a box - independent Schroedinger’s equation - Ehrenfest Theorems and applications - Basic postulates - Eigenvalues and -

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

3) Write down the time-dependent Schrödinger equation for a restricted particle.

4) Verify the correct wave functions for free and confined .

5) Solve simple problem involving the Ψ.

6) Show that the of a particle confined to a is quantized, hence draw the appropriate diagram.

7) Define eigenvalues, eigenstates and eigenfunctions

8) Explain the concept of degeneracy.

9) For trapped particles, state the features that are predicted by 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 p. In Mechanics or and courses, you have learnt that in a medium the wave disturbance in the x-direction can be represented by the transverse (y).

Recall the

 2 y   2 y   c 2   (10.1) 2  2  t  x  where c is a constant with the of .

Equation (10.1) is a . Similarly in Wave Mechanics ( a semi- constructed to solve problems involving 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 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 instead of certainties. Therefore the square of  is interpreted as the (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 about the particles is contained in complex wave functions from which probability can be got.

10.2 The Schrödinger Equation

The Schrödinger theory of specifies the laws of wave that the particles that comprise microscopic obey. (In Newtonian Mechanics the Newton’s second law, and in electromagnetic theory the Maxwell’s serve this role). This is achieved by stating, for each , 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 , 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 by   E . We can define h 2 P Wave number k   (10.3)   And E   2v  (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 it  kx for motion in the positive x-direction (10.6) and  x,t  e it  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 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    eit 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 .

If the particle is restricted the presence of a force is represented by some given 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 of energy in terms of standing waves. Just as the possible standing waves in an air column or in a stretched (see Fig. 10.1) must have an number of or half-wavelengths with the air column or stretched string so does the discrete set of and of a quantum-mechanical particle bound within a system arise.

Figure 10.1: The first four modes of of a stretched string fixed at both ends.

10.3

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 (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 will result provided A  B.

At x =0 and at x = L, ψ must be zero at all . 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)

AeiEt /   BeiEt /   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 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 ( 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 can be associated with quantum mechanical . Such observables include , 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 2t 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  nx  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  nx   n x  sin   (10.26) L  L 

2 The square of the function  n x , i.e.,  n x is the probability Pn x .

2 2 2  nx  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 or particle 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 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 , the electron probability density becomes more uniform and quantum merges with the prediction of classical .

4. The electron can escape from its trap if the trap has finite potential. This 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 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 (), momentum

, and energy respectively are given by

 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 obeys Newton’s laws. That is, he showed that if and

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 and can be associated with the classical meaning V of position and momentum of the particle and  can be identified with the classical force x component. The packet then behaves like a classical particle.

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 zz 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 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 of not more apparent in our daily experiences?

2. The wave nature of electrons has been utilized to construct an ‘’ using short-wavelength electrons to provide high resolution. (a) How might the electron beam be focused? Is it possible to construct a or 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 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 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 , described by the equation

2 in2 t /2mL2  nx   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 , 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 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.