THE SCRODINGER WAVE EQUATION

IntroductionSo far we have seen that a wave like characteristic could be associated with a material particle like electron, proton, neutron, atoms etc. In 1926, Erwin Schrodinger developed an equation for the wave associated with these elementary (small mass) particles which would describe the behaviour of the particles. The solution of the wave equation should describe the particle aspect, wave aspect and should also be consistent with the uncertainty principle.

Wave function

The quantity that characterises the de-Broglie wave is called the wave function. It is usually denoted by . It is a function of space, variables and time t. This gives complete information about the state of a physical system at a particular time. It is also called the state function and represents the probability amplitude. lf is large the probability of finding the particle is also large. lf is small then the probability of finding the particle is small. The wave function gives the likelihood of finding the particle at a given instant and at a given position.Physical significance of wave function As explained earlier, the wave function signifies the probability of finding the particle described by the wave function at the point (x, y, z) and at time t. The wave function has the following three basic properties.(i) It can interfere with itself (phenomenon of electron diffraction)

(ii) lt is large in magnitude where the particle (electron) or photon is likely to be located and small at other places. (iii) The wave function describes the behaviour of a single particle or photon and not the statistical distribution of a number of such quanta.

Probability densityThe probability of finding a particle described by the wave function is proportional to i.e., the square of the magnitude of the wave function (amplitude). We say is the probability density. If is complex then the probability density is given by *, where * is the complex conjugate of . The wave function must obey the following conditions:(i) must be finite for all values of x, y and z.(ii) must be single valued i.e., for each set of values of x, y and z, must have a unique value.

(iii) must be continuous in all regions except in those where the potential energy V(x, y, z) is infinite.

(iv) The partial derivatives of i.e., must also be continuous.Time-dependent Schrodingers equation

Consider a plane wave moving in the x-direction with a frequency , wavelength and velocity v so that and, the propagation constant. The equation of the wave can be represented by,

The energy of the wave is, according to the quantum theory and hence

Similarly the momentum,

or and

Substituting in equation of wave we get,

If is the wave function associated with a particle then the equation can be given by

Differentiating w.r.t. t,

Or

Differentiating wave equation w.r.t. x we get,

And

For a particle of mass m, moving with a velocity v; the total energy of the particle is given by

E = KE + PE = T + V

Extending this to three dimensions, the equation becomes

This is called Schrodingers time dependent three-dimensional quantum mechanical wave equation for a non-relativistic particle.

Note:1. For a free particle i.e., when the particle is not subjected to any force, the potential energy V becomeszero and the total energy is purely kinetic. The above equation becomes,

2. For a given system when we solve the Schrodingers equation, substituting the proper value of potentialenergy, we get the solutions of the differential equation which are called Eigen functions. The corresponding energy values are called Eigen b values.3. is called Hamiltonian and is represented by . The operator operated on up gives E.

Hence Schrodinger equation can be written as .

Time independent Schrodingers wave equationIn many cases the potential energy V does not depend on the time t explicitly. In these cases the forces acting on the particle and hence the potential energy depend only on the position of the particle. The wave equation for such a case can be obtained as follows:

Differentiating w.r.t. t we get,

Consider the one-dimension Schrodingers Time dependent equation,

This is Schrodingers time independent wave equation. The wave function is a function of only one variable x. Extending this to three-dimensions,

Note:

1. Schrodingers wave equation is not valid for relativistic particles since in deriving the equation the classical expression for energy and momentum are used namely and p = mv.Further, the kinetic energy = which is also a non-relativistic expression.

2. Alternative derivation of time independent equation

The wave equation of a particle is given by,

whereA is the amplitude at the point considered and independent of t but a function of(x, y, z) and is the frequency. Differentiating above equation twice w.r.t. t, we have

But, the differential equation for a wave is given by,

The total energy of the particle is given by,

E = kinetic energy + potential energy

E = + v

Extending this to three dimensions, the Schrodingers time independent equation for a non-relativistic particle

is given by,

Since this equation is independent of time, it gives the steady state form. It is very useful in systems when the energy of the particle is very small as compared to the rest energy of the particle.

Borns interpretation of the wave function An analogy between the wave function(x, t) and the location of the particle was given by Born in terms of the relation between intensity of light and the intensity of electron beam.Consider a beam of light (electromagnetic wave) incident normally on a screen. It is the result of periodically varying electric and magnetic fields perpendicular to each other and to the direction of propagation of the wave. The electric field vector is given by, E = A sin (kx t), where A is the maximum value of the electric field. We know that, the intensity I of an electromagnetic wave of frequency v is given by, where is the average of the square of the instantaneous magnitude of the electric field vector over a complete cycle, c is the velocity of light and 0 is the permittivity of free space.

If the number of photons is N, then the intensity is also given by

I = N h (per unit area per second)

Hence, = N h

This however is true only when the intensity of the beam is large i.e., N is large. The probability of striking of a photon on the screen is large at points where the intensity in the diffraction pattern is a maximum. In the case of a single photon, the probability of striking the screen is maximum at points where the wave theory predicts a maximum and lowest at place where the wave theory predicts a minimum. Thus, is a measure of the probability of the photon crossing unit area in unit time at the point under consideration. In one dimension, is a measure of the probability per unit length of finding the photon at the position x at time t. In a similar way, Max Born interpreted that the square of the amplitude of the wave function at a point is a measure of probability density (i.e., the probability per unit volume) of the particle. It is given by P = * , where * is the complex conjugate of . it should be remembered that is a complex quantity while E is real and observable. Hence, itself has no physical significance but only 2. Thus the integral of 2 over all space must be equal to 1. i.e., . Such a wave function is called a normalized wave function. The process of integration over all space to give unity is called normalization.

Particle in a one dimensional box (potential well) Consider a free particle confined to a box of length L with infinitely high walls. The particle recoils back and forth elastically as in figure. The conditions for the particle motion are,

(i) The particle moves along x-axis between x = 0 and x = L and since the recoiling is elastic, it does not lose energy during collision. Hence, the total energy remains a constant.(ii) The particle is a free particle i.e., not subjected to any external force. The potential energy is thus taken to be zero inside the box. The particle corresponding x 0 and x 0 i.e. V=0 for 0 < x < L. (The potential outside the box is infinite).

(iii) The value of the wave function outside the box is zero i.e., = O for x O and x L.

Applying the Schrodingers steady state form for one dimension,

This equation is of the form

, where

The solution is of the form

= A cos kx + B sin kx

where A and B are constants depending on the boundary condition.At x = 0, = 0

Hence,

0 = A cos 0 + B sin 0

A = 0.Again at x = L, = 0

Hence

0 = B sin kL

Since B 0,sin kL = 0

Where n = 1, 2, 3 ,

This equation gives the Eigen values of energy (proper values). The energy corresponding to n=1 is called the ground state energy. The other energies are called excited states.

To evaluate BConsider

Since the particle is inside the box, the probability of finding the particle is 1. Hence, using the normalization condition,

Hence, the Eigen functions are

, where n=1,2,3

;

;

These functions are identical with the allowed amplitude functions for the standing waves in a vibrating string fixed at both ends. The energy levels and the corresponding wave functions and the probability densities for the first three states are show.

Schrodingers equation for a particle in a three dimensional box

Consider a free particle of mass. m confined to move non- relativistically in a rectangular box of sides lx, ly, lz. The sides are parallel to the co-ordinate axes X, Y and Z. The potential energy of the particle inside the box is zero. i.e., V = 0. Hence the total energy E of the particle is equal to the kinetic energy (always a positive quantity). The Schrodingers equation becomes,

where is ,and p is the magnitude of the total momentum of the particle. is a function of x, y and z coordinates. Solving the equation we obtain,

Applying the boundary conditions namely when x = 0, x = lx , y = 0, y = ly and z = 0, z = lz and Bx = 0, By=0 and Bz=0

where A = Ax Ay Az and nx, ny, nz are independent integers. Since the particle exists inside the box, the probability of finding the particle inside the box is one.

Substituting for and solving we get,

Thus, the Eigen functions are,

The Eigen values of energy are,

Note:1. In one dimensional box we use only one quantum number n while in a three dimensional box we require, three quantum numbers one corresponding to each co-ordinate x, y, z.

2. If nx, or ny or nz is zero, = 0. This implies the absence of the particle inside the box. But this is not physically acceptable. Hence, none of the quantum numbers can be zero. The ground state corresponds nx = ny = nz = 1. Hence

The zero point energy is given by

3. For a single value of energy or momentum there are three different quantum states. The property of two or more quantum states of a particle having different sets of quantum numbers and different Eigen functions to have the same value of momentum and energy is called degeneracy. To set up the Hamiltonian for a linear harmonic oscillatorA particle which executes simple harmonic motion about at mean position is called a harmonic oscillator. At any instant, the acceleration of the particle is proportional to the A displacement and is directed towards the equilibrium position. The oscillations of a simple pendulum, the vibrations of a diatomic molecule or the vibrations of atoms in a crystal lattice are examples of harmonic oscillators.

Consider a particle executing linear simple harmonic motion. When the amplitude is small, the restoring force is proportional to the displacement and is directed towards the equilibrium position. Hence, F -x or F = - kx. The potential energy of the particle is given by,

For the particle executing simple harmonic motion

Substituting in equation

The Schrodingers equation becomes,

Let

Introducing a new variable,

For acceptable solution of the above equation, = (2n + 1) where n is an integer.Hence,

Where n = 0, 1, 2, 3 etc. These are the Eigen values of energy. The Eigen values are discrete and are given by, , etc. This shows that the energy values are quantised.Again, when n = 0, the energy is not zero. This is the zero point energy which is a consequence of the uncertainty principle.

The Eigen wave functions can be shown to be

Where and Hn (x) is the Hermite polynomial of the nth order.

The quantised energy levels of the harmonic oscillator are shown in the figure below.

Note:

1. The energy levels for the harmonic oscillator are equally spaced while in the case of particle in a boxthe energy levels diverge.

2. The wave function is given by,

When n = 0, the ground state wave function 0 is given by

The plot of the ground state wave function is shown in figure below

When n=1,

The plot of the first excited state wave function is shown in figure belowComparison of classical and quantum ideas

It has been discussed at length in the first chapter, how the classical theory fails to explain the distribution of energy in the spectrum of a black body, photoelectric effect, Compton effect, etc. and what was the need for the quantum theory. Now, we will try to bring out a comparison between the two.Quantization of energy and angular momentum According to classical ideas, a particle can have any energy continuously from zero to infinity. No restriction is laid on the amount of energy a body can have. The energy changes of radiators take place continuously. But according to Plancks hypothesis radiations are in discrete quanta called photons. A black-body radiation chamber is filled up not only with radiation but also with simple harmonic oscillators called Plancks oscillators which can vibrate with all frequencies. The energy of an oscillator of frequency vg cannot vary continuously but is limited on the discrete set of values , 2, 3, etc., where is the quantum (photon). The energy of the photon is h. This shows that energy is quantised. We have seen that the electrons in Bohr orbits have discrete energies.

According to Bohr the angular momentum of an electron is an integral multiple of. The angular momentum is quantized and given bywhere n, the principal quantum number can have integral values 1, 2, 3 etc. In 1915, W. Wilson and A. Sommerfeld discovered independently a general rule which successfully predicts the allowed energy values of periodic system. If the system has N degrees 2 of freedom with position coordinates q1, q2, q3... qn and with momentum coordinates p1, p2, p3, ..., pn then the phase integral of such a system can be defined as

According to the quantization rule only those orbits are allowed as stationary states for which each phase integral is Integral multiple of Plancks constant in a complete period.

i.e. = ni h, where ni = 1, 2, 3, etc. Again for a particle in a box, the Eigen energy is where n = 1, 2, T3, etc. i.e., the energy is quantized.

For a harmonic oscillator the energy is given by, where n = 0, 1, 2, 3, etc. The energy of a rigid rotator is where n = 1, 2, 3, etc.

These expressions for energy clearly indicate that the energy levels in these cases are not continuous but discrete and quantized. According to classical mechanics the energy can range from zero to infinity. ProbabilityIn classical mechanics, the position of an electron is clearly defined. Classically the particle is equally to be found at any point within the box for a particle in a box but according to quantum mechanics the probability of finding the particle is maximum at the antinodes and minimum at the nodes. in quantum mechanics, the probability of finding the particle is given by the wave function . The probability density is given by ||2 and is a measure of finding the particle at a particular place at a given time. In classical mechanics the position and momentum are well defined and can be determined accurately. In quantum mechanics, according to Heisenberg's uncertainty principle the position and momentum cannot be simultaneously accurately determined. If x and are the uncertainties in position and momentum then, x . The other forms of the uncertainty are in energy and time as given by E t and in angular momentum and angular displacement as given by J .

Though the calculations involving the measurement of position and momentum of bodies of large masses are remarkably accurate, the situation is different In the case of small particles such as electrons. The statements regarding the exact position and velocity of an electron are replaced by probability. In fact the Bohr concept of an atom in which the electrons revolve round the nucleus in definite orbits must be abandoned and replaced by a theory which considers the probability of finding the electron in a particular region of space at a given time.

Dual character

In classical mechanics, matter and energy are considered to be two separate entities having no relation whatsoever between the two. But we have seen, according to de Broglie a wave characteristic could be associated with a material particle. The de Broglie wavelength is given by, where m is the mass of the particle moving with a velocity v.Shortcomings of old quantum theoryThe quantum theory on Bohrs quantum condition and Wilson- Sommerfeld quantisation rule for periodic systems could explain only certain limited, problems like the energy states of hydrogen or hydrogen like atoms, particle in a box, harmonic oscillator and rigid rotator. But the main shortcomings ol the old quantum theory are:

(i) It cannot be applied to non-periodic system. (ii) It cannot explain the spectral lines of relatively simple cases like hydrogen molecule and normal helium atom.(iii) It cannot explain the relative intensities of spectral lines. (iv) The processes connected with the spin at the electrons and Pauli's exclusion principle cannot be explained.

(v) The Bohrs postulate of discrete, non-radiating energy states was empirical without supported by any theory. WORKED EXAMPLES 1. An electron is constrained in a one dimensional box of side 1 nm. Obtain the first three eigen values in eV. Solution: The eigen values of energy are given by,

For n=1

The second and third Eigen values of energy are

E2 = 22 X 0.377 = 1.508 eVand E3 = 32 X 0.377 = 3.393 eV

2. Find the momentum and energy values for an electron in a box of length 1 for n=1and n=2.Solution: Momentum is given by,

For n=1,

For n=2,

energy is given by,

For n=1,

For n=2,

E2 = 4 X 37.61 eV = 150.44 eV

3. The period of a linear harmonic- oscillator is 1 millisecond. Find its zero point energy in eV. Solution: T = 1 x 10-3 s and hence

Zero point energy

=

4. The energy of a linear harmonic oscillator in its third excited state is 0.1 eV. Calculate its frequency. Solution:

5. A proton is confined to a nucleus of radius 10 -14 m. Calculate the lowest energy and also the next two energy Eigen values of the proton given mp = 1 .67 x 10-27 kg.Solution:

For n=1

The second and third Eigen values of energy are

E2 = 22 X 2.05 = 8.2eV

and E3 = 32 X 2.05 = 18.45 eV6. Calculate the zero point energy and the spacing of the energy levels in a one dimensional oscillator with an oscillator frequency of 1 kHz.Solution:

Hence, spacing of energy levels, E = E1 - E0 = 4.146 X 10-12 eV 7. An eigen value of an electron confined to a one dimensional box of length 0.2 nm is 151 eV. What is the order of the excited state? Solution:

8. Find the probability that a particle in a one dimensional box of length L can be found between 0.4 L to 0.6 L for the ground state.Solution: The wave function in one dimensional box of length L for a free particle is given by

The probability of finding a particle in a distance x about the mean position x isPn x = n n* x

Mean position,

And x = 0.6 L 0.4 L = 0.2 LFor the ground state, n = 1

EXERCISES I Questions1. Obtain the time independent Schrodingers equation. Give its physical significance.2. Obtain Schrodingers time-dependent equation for a free particle. 3. Write Schrodingers wave equation tor a free particle. What is the physical interpretation of the wave function ? Explain Borns interpretation.4. Explain the terms eigen values and eigen functions. Give the physical significance of a wave function. 5. Explain the requirements that are imposed on a physically acceptable wave function.6. Obtain the Schrodingers equation for a particle in a one dimensional box and solve it to obtain the energy eigen values. Also represent the first three wave functions in a graph.7. Write down the Schrodingers equation for a particle in a one dimensional box. Show that the linear momentum and energy of a particle in a one dimensional box is quantised.

8. Obtain Schrodingers equation for a linear harmonic oscillator. Solve it to obtain the energy eigen values. 9. Show that the energy of a harmonic oscillator is quantised in steps of h. Explain the existence of zero point energy.10. Bring out a comparison between the classical and quantum ideas with reference to the quantisation of energy and angular momentum. 11. What is the basic difference between the postulates of classical and quantum theory? II Concept questions12. Distinguish between a free particle and particle in a box. 13. The zero point energy of a harmonic oscillator is not zero. Explain.

14. A bound particle has quantised energy values whereas it is not so for a completely free particle. Explain. 15. The concept of trajectory has no significance in quantum mechanics. Explain.16. Bring out the physical significance. of zero point energy. 17. Explain the concept of Bohrs stationary orbits on the basis of quantum mechanical model of the atom. EMBED Equation.DSMT4

EMBED Equation.DSMT4

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