Prof. J. K. Goswamy�UIET, Panjab University�Chandigarh

1

Applications of Schrodinger Equation�

• Quantum Mechanics of a Free Particle.
• Particle in Infinitely Rigid Box.
• Particle Incident on Potential Step.
• Particle Incident on Finite Potential Barrier.
• Tunnel Effect and its Applications.
• Particle in Harmonic Oscillator Potential Well.

2

Using Schrodinger’s Equation

Quantum Mechanics of Free Particle

Free Particle

• Let’s consider that a particle of mass m and kinetic energy E is moving in free space i.e. V(x)=0. The Schrodinger’s equation for this particle is:

4

5

• If the particle is present somewhere in the universe, then

​

​

​

• The particle has equal probability of being present everywhere in universe. Hence the particle is delocalized.
• The integral is infinitely large and the normalization condition is satisfied only if ΙAΙ² is infinitely small.

​

​

6

The delocalization of the particle and the normalization problem of the wave function are correlated to each other.

7

• Conclusions:
• The pure harmonic wave can’t represent a free particle.
• Rather a wave-packet obtained by superposition of harmonic waves of different wavelengths is a correct representation.
• The wave function of the free particle is a wave packet (or wave group) propagating with same speed as the particle.

Potential in Quantum Mechanics

• Quantum Mechanics studies dynamics of the microscopic system.
• The force is inserted in Schrodinger equation as the potential distribution created by it in space and time.
• The system driven under the influence of repulsive force has +ve potential distribution referred to as potential barrier.
• The system bound by attractive forces has -ve potential distribution forming the so called potential well.
• The solution of Schrodinger equation leads to allowed wave functions and energies characterizing the microscopic system.

8

Some Potential Distributions

​

​

• Potential distribution at pn junction owing to different doping of p- and n-type wafers.

​

• Electrostatic potential distribution due to a nucleus which binds the electrons in the atom.

​

​

• Attractive potential distribution due to a nucleus which binds the nucleons within the nuclear volume.

X=0

Potential step

X=0

X=L

Potential barrier

Potential well

-V_o

V_o

9

Particle in Infinitely Rigid Box

Particle in Infinitely Rigid Box

• Let’s consider a particle of mass m trapped in infinitely deep potential well or an infinitely rigid box as shown:

​

11

• This particle is constrained to be present in the region 0<x<L as it requires infinite energy to be in either of the regions x<0 or X>L.
• The Schrodinger’s equation applied to the region 0<x<L of the potential well is

​

​

12

The solution of this differential equation is:

​

​

Since the particle can’t be found outside the potential well so wave function must obey boundary conditions:

13

14

Quantization of Energy

15

Allowed Wave Functions

• The normalized wave function, whose absolute square is probability of particle at a position x in the box, is

16

Discrete States

E₁

E₂

E₃

17

Conclusions

• Particle trapped in a rigid box represents a situation of tightly bound electrons in atom or highly bound nucleons in nucleus.
• A bound particle can possess only certain quantized energies.
• The state of particle is defined by quantized energy and the allowed wave function.
• Probability distribution of bound particle varies in different states.
• Zero energy is prohibited for a bound particle as it violates the Heisenberg’s uncertainty principle.
• If the potential at the surface of the well is taken to be zero then all the states will be of negative energy which is the characteristic of a bound system.

Particle Incident on Potential Step

Particle Incident on Potential Step with E>V_o

• Let’s consider that particle of mass m and kinetic energy E is incident on a one dimensional potential step defined as:

​

​

​

​

​

​

​

​

20

The kinetic energy of particle is greater than the height of potential step (i.e. E > V_o).

• The steady state Schrodinger’s eqn. for two regions of potential are:

​

21

• The first term in eq. (2.92) represents the wave incident on the potential step.
• The second term in eq. (2.92) represents the reflected wave at x=0 interface.
• The first term in eq. (2.93) is wave transmitted through the potential step.
• The coefficient D vanishes as no further reflection possible in the region x>0.

• The constants A, B and C can be evaluated using the boundary conditions on wave-functions at x=0. Hence we have:

​

​

​

​

​

Solving eqns. (2.94) and (2.95), we get:

​

22

• The evaluated coefficients B and C can be inserted in terms of eqs. (2.92) and (2.93) to obtain the final form of incident, reflected and transmitted matter waves. Hence we have:

​

23

• Reflection Coefficient at the potential discontinuity is defined as the ratio of intensity of reflected wave to the intensity of incident wave. Hence:

​

​

• Transmission Coefficient at the potential discontinuity is defined as the ratio of intensity of transmitted wave to the intensity of incident wave. Hence:

​

​

24

Conclusions

It can be observed that R+T=1 which implies that when a particle is incident with E>V_o, on the potential step, it has the probability of getting reflected as well as transmitted. This is in contradiction to predictions of classical mechanics where such a particle is expected to be solely transmitted.

25

Particle Incident on Potential Step with E<V_o

• Let’s consider that a particle of mass m is incident on a one dimensional potential step with its kinetic energy E<V_o.
• The Schrodinger’s equation for two regions are:

26

• The coefficient D=0 as argued in previous case. The constants A, B and C can be evaluated using the boundary conditions on wave-functions at x=0. Hence we have:

​

​

​

​

27

• The evaluated coefficients B and C can be inserted in eqn. (2.108) to obtain the final form of incident, reflected and transmitted matter waves. Hence we have:

​

​

​

​

​

​

​

​

28

• Reflection Coefficient at the potential discontinuity is:

​

​

​

​

​

​

​

When particle of energy less than height of potential step is incident, it is totally reflected without any transmission through the potential discontinuity. This agrees with the prediction of classical mechanics.

​

​

​

​

29

Particle Incident on Potential Barrier

Finite Potential Barrier

• Let’s consider a particle of mass m and kinetic energy E incident on a potential barrier. The potential distribution of this barrier is expressed as:

​

​

​

​

​

​

​

• The energy of incident particle is less than height of potential barrier.

​

​

​

31

• The Schrodinger’s equation for three different regions of potential barrier are expressed as:

​

32

​

​

​

​

​

• The solutions for differential equations (2.113-2.115) are given as:

​

​

​

• The A, B, C, D, F and G are arbitrary constants to be determined from the boundary conditions on wave functions at two potential discontinuities located at x=0 and x=L.

33

• In equations (2.116), the first term represents the wave incident on the potential barrier (at x=0) while second one represents the reflected component.
• In equation (2.117), the first term represents the portion of incident wave transmitted through the x=0 barrier while the second term represents the component of wave reflected at boundary x=L. Both of these waves are damped in nature.
• In equation (2.118), the first term represents the portion of incident wave transmitted through the x=L barrier.
• Since there is nothing to reflect a wave in the x>L region, so second term must be non-existent (G=0). Thus equation (2.118) takes the final form as:

34

• The wave-function given by equations (2.116) and (2.117) must satisfy boundary conditions of being single valued at x=0, where potential changes abruptly. Hence we get:

​

​

• The wave-function must be continuous at x=0. Hence we get:

​

​

​

​

​

​

35

• Applying boundary conditions for equations (2.117) and (2.118) at x=L, we get:

​

​

​

​

​

• C and D coefficients can be determined in terms of coefficient F from equations (2.122) and (2.123) as:

​

​

36

• Putting values of C and D in equations (2.120) and (2.121), we can evaluate A and B as:

37

• The coefficient B defines the amplitude of wave, that is reflected at potential barrier (at x=0). The reflection coefficient is expressed as:

​

​

​

• The transmission coefficient through x=L boundary of the barrier is:

​

​

​

• The probability of transmission is not zero indicating that a particle moving with energy E<V₀ will penetrate (or Tunnel) through the barrier. This is in contradiction to the expectation of classical mechanics and this is called barrier penetration or tunnel effect.

​

​

​

38

Tunneling Effect and its Applications

What if the Box is Finitely Rigid ?

• If box is finitely rigid then probability of trapped particle being found outside the box is finite although small.
• The energy states of such a particle get depressed due to increased de-Broglie wavelength of the particle.

​

​

​

• The particle, with energy lesser than depth of well, has finite although small probability of being found outside the well too. This is due to leakage of the particle through the wall. This phenomenon is called the Tunneling effect.

40

Tunneling Effect

• When a microscopic particle is trapped in a potential well such that its energy is less than depth of the well.

or

• When the particle is incident on a potential barrier such that its energy is less than height of the barrier.
• Classically particle will not be able to come out of potential well or cross the barrier.
• Quantum mechanically this particle will have finite though small probability of crossing the well or barrier by leaking through the wall. This is Tunneling effect.

41

• The property of the barrier penetration or tunneling is a consequence of the wave nature associated with the moving quantum mechanical particle.
• It is analogous to a wave which is incident on interface of two media and it suffers reflection as well as transmission.
• The tunneling probability for a particle of kinetic energy E through a barrier of height V_o and width L is given as:

​

​

• This probability is the ratio of particle flux penetrated to that incident on the barrier.
• The tunneling probability
• decreases with increase in width of the barrier.
• Increases when the quantity (V₀-E) decreases.

​

42

Uncertainty Principles and Tunneling

Barrier Penetration & Heisenberg’s Uncertainty Principle

• For a particle trapped in potential well, its probability outside the well is zero. Hence we have

​

• The above results implies that such a trapped particle will infinite momentum and energy and hence can always come out of the well or overcome the barrier.
• Consequently the probability of the particle outside the well or on other side of barrier can never be zero.

Barrier penetration & Time-Energy Uncertainty Principle

• A particle with kinetic energy less than the barrier height is incident on the potential barrier.
• The time of interaction of the particle with potential barrier is very small.
• Due to small interaction time, its energy appears to be associated with very large uncertainty.
• The particle apparently has energy more than the barrier height for a short duration equal to interaction time.
• The particle crosses the barrier in this short duration.

​

44

Alpha Emission-Geiger Nuttall Rule

• The α-particles are emitted by heavy mass nuclei (Ra, Rn, U, Pu, Th) to achieve stability against radioactive decay.
• Geiger and Nuttal made a systematic study of experimental values of half-lives of α-emitters and kinetic energy of emitted α-particles. This led to empirical Geiger-Nuttal rule which states that:
• Long-lived radionuclides emit low energy α-particles
• Short-lived radionuclides emit energetic α-particles.

45

• If α-particle is incident on heavy nuclei (Ra, Rn, U, Pu, Th) then it has to overcome a coulomb barrier of height ~ 25-30MeV before it penetrates into the nucleus.
• Hence incident α-particle must have kinetic energy more than coulomb barrier height.
• Expectedly during the α-decay from such nuclei, the emitted α-particle must have kinetic energy of the same magnitude as is required to cause their penetration into nucleus.
• However it is observed that α-particle emitted from such nuclei have energy 4-9 MeV which is much lesser than predicted value of 25-30MeV.

​

​

46

• The process of α-particle emission was explained by Gammow in 1928 using the concept quantum mechanical tunneling.
• This is explained by the fact that α-particle is formed inside the nucleus just before decay.
• The formed α-particle starts oscillating between two walls of the well at very high frequency.
• During each collision with nuclear wall, there is small but finite probability of particle’s leakage which causes emission of α-particle.

The tunneling probability is greater for energetic α-particles and hence lifetime of parent nuclide is small. This justifies the Geiger-Nuttall rule.

Field Ionization

• If an electric field, of sufficient strength, is applied across the ends of an insulator, then electrons can break away from their parent atoms.
• This is a property which is expected classically too.
• However it is observed that low values of applied electric field also causes electron removal, referred to as Field ionization.
• In such cases, the ionization of electrons occurs due to tunneling of bound electrons through the potential well created by electrostatic field of nucleus.

​

48

Tunnel Diode

• The working, efficiency and mode of operation of crystal diodes can be greatly improved by sandwiching an insulating layer between p- and n-type crystals.
• Such diodes are called Tunnel diodes and current in them flows due to tunneling of potential barrier caused by mid-layer of insulator.

​

49

Ammonia Inversion

• In a molecule of ammonia, the Coulomb’s forces of repulsion exerted by three similar hydrogen atoms on one another produces a kind of potential barrier, which prohibits the nitrogen atom from moving from one end on the axis of symmetry to the other end.
• Classical mechanics demands that a large amount of energy is needed to be supplied to nitrogen atom to shift from one to other end position.
• However this change in position occurs 10 billion times per second due to tunneling through the barrier created by three hydrogen atoms.

​

50

Particle in Simple Harmonic Oscillator Potential

Characteristics of SHM

• Harmonic motion is the vibration of system about unique equilibrium configuration.
• Essential conditions for oscillating system are:
• Restoring force.
• Inertia of the system.
• Differential equation for harmonic motion is

​

52

• Potential energy of harmonic oscillator at displacement x from equilibrium position is:

​

• The Harmonic oscillator potential well is characterized by particle energies less than the height (V_o) of the well.
• Examples of microscopic oscillators:
• Lattice vibrations in crystal.
• Vibrations in a molecule.
• Nuclear surface Vibrations.

​

53

Predictions of Classical Theory

Classical theory of simple harmonic oscillators predict the following:

• The simple harmonic oscillator oscillates strictly between the classical turning points defined by x=+A and x= -A as its kinetic energy tends to be negative if it moves into the region
• The total energy of the oscillator is proportional to the square of the amplitude and can have any value.
• The probability of finding the oscillator is minimum at mean position (velocity is maximum) and is maximum at end positions (velocity is minimum).

54

Quantum Mechanical Treatment

• If a particle of mass m executes motion in one dimensional harmonic oscillator potential well, the Schrodinger eq. is:

​

​

​

​

​

​

​

​

• To Solve diff. eq. (2.131), quantities α and β are introduced:

​

​

​

​

​

​

55

• Using eqs. (2.132,2.133) in eq. (2.131), transformed Schrodinger equation is:

​

• Let’s change the variable of description x by the substitution

​

​

​

​

• Putting (2.135) in (2.134) and solving, we get:

​

​

​

​

​

​

​

​

​

56

Asymptotic form of Wave Function

• Now ψ is a function of y and to represent a wavefunction of a particle localized in space, it must be finite everywhere and vanish as y→∞. For asymptotic solution we can assume that y>>β/α. Hence we have:

​

• The asymptotic form of wave function, satisfying the limiting condition given by differential equation (2.137), is expressed as:

​

Hence total wave function forming the solution of differential equation (2.136) will be

​

​

​

​

​

​

​

57

58

Inserting the wave function given by (2.138) in eq. (2.136), we get:

​

​

​

​

​

​

​

The solution of differential equation (2.139) gives the function H(y). Equation (2.139) is in most appropriate form with substitution:

​

​

Substituting (2.140) in (2.139), we get:

​

• Using the power series method to obtain solution of diff. eq. (2.141):

​

​

​

​

​

​

​

• For wave function to vanish at infinity, the component H(y) must have finite terms instead of being a infinite series. Hence if the function H(y) has first n terms non-vanishing, then from the result (2.145), we have:

​

​

59

Energy of Harmonic Oscillator

• Using the recursion formula and considering that H(y) has first n terms which are non-vanishing, the energy in different states of harmonic oscillator is

​

​

​

​

• The lowest energy state has non-zero energy, called zero-point energy.

​

​

60

61

Wave Function of Harmonic Oscillator

• The solution H(y) of diff. eq. (2.141) on truncation results in Hermite polynomial. The total normalized wave function is expressed as:

​

​

• Hermite polynomials of first few orders are given below:

​

62

• The wave function of harmonic oscillators in first few states are shown below:

​

63

Conclusions

• The quantum mechanical oscillator penetrates or tunnels into otherwise classically forbidden regions beyond x=±A.
• The probability distribution of position of a harmonic oscillator differs in various states of harmonic oscillator.
• In the lowest state, the quantum mechanical oscillator has most probable position at centre of oscillator potential well but classically it is expected to have maximum velocity and (hence minimum probability) at equilibrium position.
• The most probable position of quantum oscillator is same as predicted classically for states with large values of n (Bohr’s Correspondence principle).

​

64