Quantum Mechanics - drmgriyearbtech.com · 2020. 2. 10. · Davisson - Germer Experiment Experimental arrangement In 1927, Davisson and Germer for the first time proved the wave nature

1.1 QUANTUM FREE ELECTRON THEORYFollowing de Broglie’s hypothesis of matter waves and Schroedinger’s matter

wave equaion, Sommerfeld proposed the quantum free electron theory. He assumedthat the valence electrons are free in ametal piece and they are moving in an uniformpotential within the metal. He assumed that a moving electron behaves as if it is asystem of waves, called matter waves. By this theory Sommerfeld correctly predictedthe values of specific heat of metals, electrical conductivity and thermal conductivity ofmetals.

1.2 DE BROGILE WAVESA wave is spread out over a relatively large region of space and it cannot be said

to located just here and there. Actually a wave is nothing but rather a spread outdisturbance. A wave is specified by its frequency, wavelength, phase or wave velocity,amplitude and intensity.

A particle (matter) has mass and it is located at some definite point. It can movefrom one place to another and it gives energy when slowed down or stopped. Theparticle is specified by its mass, velocity, momentum and energy.

Considering the above facts it appears difficult to accept the conflicting ideasthat radiation has a dual nature. That is, radiation is a wave which is spread out spaceand also a particle which is localised at a point in space.

However this acceptance is essential because the experimental results for blackbody radiation, photoelectric effect and X-ray absorption are explained by consideringthe radiation to appear as a stream of particles.

Quantum Mechanics1

1.2 ENGINEERING PHYSICS - II

The success of Planck’s interpretation of black body radiation, Einstein’sinterpretation of photoelectric effect, Bohr’s interpretation of the line spectra, Compton’sinterpretation of the scattering of X-rays, etc., convinced everyone of the photoncharacter of radiant energy. On the other hand, electromagnetic radiation, which includesvisible light, infrared and ultraviolet radiation, X-rays and gamma rays was shown to bea wave motion by interference and diffraction experiments. Radiation thus exhibits adual wave-practicle nature.

One can use either the wave aspect or the particle aspect of light to explain theabove observed phenomena as it suited them. Here it should be remembered that radiationlike light cannot exhibit its particle and wave properties simultaneously.

de Broglie waves or Matter waves: The waves associated with the particles ofmatter (e.g. Electrons, protons, etc.,) are known as matter waves or pilot waves or deBrogile waves.

1.3 de BROGLIE’S HYPOTHESIS

In 1924, Louis de-Broglie made a daring suggestion that like radiations matteralso exhibits dual characteristic. In other words, particles of matter like electrons andprotons also exhibit wave properties. de Broglie’s suggestions were based on thefollowing facts :

1. The entire universe consists of matter and radiation (energy) only.

2. Nature is symmetrical in so many respects. Therefore the two physical entitiesviz matter and energy must be mutually symmetrical. That is to say if radiantenergy has dual characteristic, matter must also have dual (i.e. particle likeand wave like) nature.

So a moving particle is associated with a wave which is known as matter waveand its wavelength is given by

ph

mvh

Where m is the mass of the particle; v is its velocity and p is its momentum.

Derivation of de Broglie equation

The expression of the wavelength associated with a material particle can bederived on the analogy of radiation as follows :

QUANTUM MECHANICS 1.3

Considering the Planck’s theory of radiation, the energy of a photon is given by

)aspectwave(hChvE

Where C is the velocity of light in vacuum and is its wavelength. If the photonis considered as particle of mass m then according to Einstein’s mass - energy relation,the energy of the photon is given by

E = mC2 (particle aspect)

Since the energy of the photon in the two cases is the same, therefore fromequations (A) and (B) we get

2mChC

(or) mCh

mChC

2

The quantity mC = p, the momentum of the photon and therefore

ph

de Broglie carried over this idea to material particles. Thus if a particle has amass m and travels with a velocity v, its momentum is mv and the wavelength associ-ated with this particle is given by

h hλ = =mv p

de Broglie’s wavelength associated with electrons

Let us consider the case of an electron of mass m and charge e, accelerated bya potential V volt from rest to velocity v. Then,

Energy gained by electron = eV

Kinetic energy of electron = 2mv21

eV = 2mv21


v =meV2

NoweVm2h

meV2m

hmvh

Substituting h = 6.6256 × 1034 J-s,

e = 1.602 × 10-19 C

and m = 9.11 × 10-31 kg.,

We get 2/13119

34

)1011.9V10602.12(106256.6

= V26.12

Vm1026.12 10

Å

So when an electron accelerated through a potential difference of 100 volt, thede Broglie’s wavelength associated with it is 1.226 Å

Properties of matter waves

1. Lighter is the particle, greater is the wavelength associated with it.

2. Smaller is the velocity of the particle, greater is the wavelength associatedwith it.

3. These waves are not electromagnetic waves.

4. Matter waves are generated by the motion of particles. If the particles are atrest, then there is no meaning of matter waves associated with them.

5. The wavelength of matter waves are independent of charges on the particles,but depends upon the velocity of particles.

6. The wave velocity of matter wave can be greater than the velocity of light.


The phase or wave velocity of the matter wave is u = f where f is the frequencyof matter wave. Now, by photon analogy, the energy of the material particle,

E = h f (or) f = hE

Also energy of the particle by Einstein’s relation

E = mC2 h

mCf2

Now mvh

vC

mvh

hCfu

22

Since the particle velocity ‘v’ cannot exceed velocity of light (secondpostulate of special theory of Relativity) the phase velocity of matter wave isgreater than the velocity of light].

7. The wave nature of matter introduces an uncertainty in the location of theposition of the particle because a wave cannot be said exactly at this point orexactly at that point. However where the wave intensity is large (strong) thereis a good chance of finding the particle while where the wave intensity is small(weak) there is very small chance of finding the particle.

8. They are pilot waves in the sense that their only function is to pilot or guide thematerial particles.

9. The matter wave is not a physical phenomenon. It is rather a symbolicrepresentation of what we know about the particle. It is wave of probability.

1.4 EXPERIMENTAL VERIFICATION OF MATTER WAVES

Davisson - Germer ExperimentExperimental arrangement

In 1927, Davisson and Germer for the first time proved the wave nature ofelectorns. The basis of their experiment was that since the wavelength of an electron isof the order of X-rays, a beam of electrons must show diffraction effects from a crystal,like X-rays.


(a) Experimental (b) Scattering Curve (c) Determination of

arrangement Bragg’s angle ‘’

Figure 1.1

Fig.1.1(a) shows their experimental arrangement. The electorns from a hotfilament ‘F’ are accelerated through a suitable potential difference. The narrow beamof electrons emerging from opening P is made to strike the face of single nickel crystalperpendicularly.

The electrons are scattered in all directions by the atoms of Ni crystal. The intensityof electron beam scattered (i.e. number of scattered electrons) in a given direction ismeasured by allowing it to enter an electron detector. The electron detector is so arrangedthat it can be rotated around the nickel crystal through a large angle. The output of thedetector is measured by a sensitive galvanometer. The deflection of the galvanometeris directly proportional to the intensity of beam entering the detector.

Observations

i) The electrons are scattered in all directions showing that they are acting likewaves. If the electrons were just particles, they would have been simplyreflected straight back.

ii) The intensity of scattered electron beam depends upon the angle ‘’ andaccelerating potential ‘V’ (Figure 1.1(b)). It was observed that high intensityof scattered electron beam always occurred at an angle = 50o for 54 Vaccelerating voltage. Here ‘’ is the angle between the incident and scatteredbeam of electrons with respect to top surface of the crystal.


Theory

The atomic rows of nickel crystal act like the rulings of a diffraction grating.Under the above conditions (V = 54 volt, = 50o), the crystal is producing the firstorder Bragg reflection at an angle = (90o - /2) where is called angle of diffractionwhich is the angle between the incident beam and the crystal plane (Figure 1.1(c)). Thewavelength of the electron wave can be calculated from the Bragg’s formula asunder:

n = 2d sin

For first order reflection n = 1, d = a sin 25o = interplanar distance

For nickel, interatomic distance a = 2.5 × 1010 m

d = 2.15 × 1010 sin 25o = 0.909 × 1010 m

sin = sin (90o 25o) = sin 65o

= 2 × 0.909 × 1010 × sin 65o = 1.65 × 1010 m

(or) = 1.65 Å

The theoretical value of electron wavelength can be calculated using de Broglie’srelation.

67.15426.12

Å

Thus there is an excellent agreement between the experimental value and thetheoretical value of wavelength of electron wave. This confirms the de Broglie’shypothesis of matter waves.

1.5 UNCERTAINTY PRINCIPLEIn 1927, Heisenberg proposed uncertainty principle which is a direct consequence

of the dual nature of matter. It states that in any simultaneous and accuratedetermination of the values of both members of particular pairs of physical variablesthat describe the behaviour of an atomic system, the product of the uncertainities

(or) errors in the knowledge of two variables is equal to or greater than 4πh

where

h is the Planck’s constant.


Considering the pair of physical variables of a particle as position and momentum,we have

px x > h / 4

where px is the error (or) uncertainity in determining the momentum along x axis andx is the uncertainity in determining the position of the particle. Similarly

E t > h/4

J > h/4

where E and t are the uncertainties in determining the energy and time while J and are the uncertainties in determining the angular momentum and angle.

Special Features

According to the uncertainty principle it is not possible to determineaccurately and simultaneously the values of position and momentum of aparticle at any time. If we want to determine the position of an electronvery accurately, then this can only be done at the expense of accuracy indetermining its momentum and the error involved in momentum measurementwill increase.

Thus uncertainty principle sets limits on measurement accuracy in the case ofatomic particles only and it sets no limit for heavy objects whatever on ourmeasuring procedures.

The uncertainty relation shows us why it is possible for both light and matterto have a dual, wave-particle, nature. It is because these two views so obviouslyopposite to each other, can never be brought face to face in the sameexperimental situation.

Matter and light are like two faces of a coin that can be made to display eitherface at will but not both faces simultaneously.

Uncertainty principle helps us to understand several phenomena related tosub-atomic particles. It reveals that electrons cannot exist inside the atomicnucleus.

1.6 ELECTRON MICROSCOPEElectron microscope uses the electron beam from electron gun as source of

radiation. Further the objective and projector lens are made from magnetic currentcarrying coils. The focal length of the magnetic lenses are variable. There is one limitationon the electron microscope such that it can work under vacuum. Any how thay havehigh resolving power (resolution limit 2 Å) and high magnification power (106).


There are three principle binds kinds of electron microscopes : (a) Transmissionelectron microscope (TEM) (b) Scanning electron microscope (SEM) and (c) Fieldemission electron microscope.

In the first two types, electron gun gives a electron beam which acts as thesource of radiation. In the last filed emission type, the specimen itself is a source ofradiation.

In the modern times, TEM is not widely used. Since the living objects can not bestudied due to vacuum. Further the preparation of replica of the specimens is acumbersome.

It is a type of microscope in which instead of light beam, a beam of electrons areused to form a large image of a very small object. These microscopes ae widely usedin the field of medicine.

Principle

A steam of electrons are passed through the object and the electrons whichcarries the information about the object are focussed by electric and magneticfields.

Since the resolving power is inversely proportional to the wavelength the electronmicroscope has high resolving power because of the shorter wavelength (103 timesshorter than the wavelength of the visible light.

Construction

Electron gun

C1

C2

Iron case

Figure 1.2


An electron microscope is similar to that of an optical microscope. Here thefocussing of electrons can be done either by magnetic lens (or) electrostatic lens.Normally in electron microscope magnetic lenses are used for focussing.

In general the magnetic lenses are made of two coils C1 and C2 enclosed insidethe iron cases, which have one hole each as shown in Figure 1.2.

When the holes faces each other, the magnetic field in the space between thetwo coils focus the electrons emerging out from the electron gun. Similarly the divergenceof the electrons can also be made by adjusting the position of the holes in the iron cases.

B A

A1 B1Real image

Slit

Magnetic Condensing lens

Magnetic Object lensObject

Magnetic Projector lens

B2 A2

Final image

Electron gun

Vacuum chamber

Figure 1.3

The essential parts of an electron microscope are as shown in Figure 1.3 and forcomparison an optical microscope is also shown aside in Figure 1.4. It consists of anelectron gun to produce, steam of electrons. Similar to condensing lens, objective andeye piece in an optical microscope here three magnetic lenses are used, viz.

i) Condensing lens to condense the electron.

ii) Objective lens to resolve the structre of the specimen and

iii) A projector lens, similar to eye piece for enlargement.


B A

A1 B1(Real image)

Light source

SlitCondensing lens

Object lensObject

Intermediate image

Projector lens(Eye piece)

B2 A2

Final image (Virtual image)

Figure 1.4

The whole arrangement is kept inside a vacuum chamber to allow the passge ofelectron beam.

Working

Steam of electrons are produced and accelerated by the electron gas. The electronbeam is made to pass through the centre of the doughnut shaped magnetic condensinglens. These electrons are amde as parallel beam and is focussed onto the object AB(Figure 1.3). The electrons are transmitted more in the less denser region of the objectand is transmitted less (i.e) absorbed by the more denser region of the object. Thus thetransmitted electron beam on falling over the magnetic objective lens, resolves thestructure of the object to form a magnified real image of the object. Further the imagecan be magnified by the magnetic projector lens and the final image is obtained on afluorescent screen.

In order to make a permanent record of the image of the object, the final image isalso be obtained on a photographic plate.


Advantages

i) It an produce magnifications as high as 1,00,000 times as that of the size ofthe object.

ii) The focal length of the microscopic system can be varied.

Applications

It has a very wide area of applications (e.g) in biology, metallurgy, physics,chemistry, medicine, engineering, etc.

i) It is used in determine the complicated structure of the cyrstals.

ii) It is used in the study of colloids.

iii) In industries it is used to study the structure of textile, fibres, surface of metals,composition of paper, paints, etc.

iv) In the medical field it is used to study about the structure of virus, bacteria,etc., which are of smaller size.

1.7 SCANNING ELECTRON MICROSCOPE (SEM)The surface structure or topography can be studied easily in the scanning electron

microscope than in the transmission electron microscope. The depth of focus of thescanning electron microscope is so high that a fracture surface can be directly examinedwithout any polishing.

Principle

Here the image of the surface of the specimen is builtup by using an electronprobe of very small diameter which scans the surface of the specimen and collectingthe emitted secondary electrons or scattered electrons or transmitted electrons.

Construction

A high energy electron beam is produced by an electron gun (figure 1.5). Thebeam is focussed to a spot of about 100Å diameter and made to scan the surface of thespecimen. The scintillator collects the secondary or scattered electrons and convertsinto light signal.

The light signal is further amplified by photomultiplier and that signal is used tomodulate the brightness of an oscilloscope spot which transverse a raster in exactsyncronism with the electron beam at the specimen surface. The image observed onthe oscilloscope screen is similar to the optical image. The specimen is usually titledtowards the collector at a low angle (<30°) to the horizontal, for general viewing.


Image

Scangenerator

Photomultiplier

Collector

Magnificationcontrol

Objective lens

Scanningcoils

Specimen

2 condenser lensnd

1 condenser lensst

Anode

GridElectron gun

Filament

Figure 1.5: Scanning electron microscope

Working

The high energy electron beam (30kV) is incident on the specimen surface, sowe can get back scattered high energy electrons and emitted low energy secondaryelectrons (100 eV). Since the secondary electrons are of low energy they can be bentround corners and give rise to the topographic contrast.

The intensity of backscattered electrons is proportional the atomic number andthese electrons are not so easily collected by the normal collector. If the secondaryelectrons are to be collected a positive bias of 200 V is applied to the detector.

These electrons produce scintillations of light. Then it is amplified by photomultiplierunit and is finally given to the grid of the cathode ray tube. The scanning system is insynchronous with the scanning frequency of the specimen. Therefore during the scanningof the fluorescent screen an optical image is created which contains the topographicalfeatures of the surface of the specimen.

Advantages

i) It can used to examine specimens of large thickness.

ii) It has large depth of focus.

iii) It can be used to get a three dimensional image of the object.


iv) Since the image can be directly viewed in the screen, structural details can beresolved in a precise manner.

v) The magnification may be upto 300,000 times greater than that of the size ofthe object.

Disadvantages

i) The resolution of the image is limited to about 10-20 nm, hence it is very poor.

Applications

i) It is used to examine the structure of very large specimens in a three dimensionalview.

ii) Similar to the applications of electron microscope this scanning electronmicroscope also has applications over various fields such as Biology, Industries,Engineering, Physics, Chemistry, etc.

1.8 PHYSICAL SIGNIFICANCE OF THE WAVE FUNCTION ‘’1. The wave function ‘’ measures the variations of the matter wave. Thus it

connects the particle and its associated wave statistically. It is the complexamplitude of the matter wave.

2. The wave function or complex displacement is a complex quantity and wecannot measure it.

3. The wave function is used to identify the state of a particle in an atomicstructure.

4. It can tell the probability of the position of the particle at a time but it cannotpredict the exact location of the particle at that time. Thus it tells us wherethe particle is likely to be not where it is.

5. We can say that the wave function as probability amplitude since it is used tofind the location of the particle.

6. The probability of finding a particle in a particular volume element d is givenby

P(r) d = |*| d = | (r, t) |² d

where * is called the complex conjugate of Here | |² is proportional toprobability density of the particle in the state . A large value of it indicatesthe high probability of the particle’s presence while a small value of it meansthe small or low probability of its presence.


7. | |² d = 1 when the particle’s presence is certain in the space.

8. Being a complex function, it does not have a direct physical meaning, butwhen we multiply this with its complex conjugate, the product | |² has thephysical meaning. i.e. we will speak normally the intensity of light at a pointrather than the amplitude of light at a point since the intensity (square of theamplitude) is a measurable and real quantity. The same thing is true for matterwaves also.

1.9 SCHROEDINGER’S WAVE EQUATIONThe Schroedinger wave equation, like Newton’s laws of motion is a fundamental

equation in quantum mechanics to explain the various phenomena associated with theatoms and molecules. Newton’s laws of motion can be applied only to macroscopicsystems and events. But the Schroedinger wave equation can be applied both tomacroscopic and microscopic systems and events. Here the wave nature of electronsis taken into account. It is a matter wave equation. Using this equation we can get theallowed values or eigen values of energy of electron in an atom or molecule or metal.

The classical wave equation

2

22

2

2

xyv

ty

(in one dimension) .... (1)

where y is the displacement of the particle, which is moving in x direction, at any instant‘t’. This equation can be applied to waves in a stretched string, sound waves in air andlight waves in vacuum. The general solution of this equation is equal to

y (x, t) = A exp (–i )

vx–t .... (2)

= A cos (t – x/v) – iA sin (t – x/v)

Only the real part of this equation has significance in the case of stretched stringand so the imaginary part is discorded as irrelevant.

According to Schroedinger, for an atomic particle like electron, one must take thewhole solution of y, since one cannot determine the momentum and position of itsimultaneously. He called this complex displacement as wave function ‘’. Since it is acomplex, we cannot measure it.


So we specify by

= A exp x– i t –v

= A exp x–2 i t –v

(or) = A exp x–2 i t –

since v = .... (3)

The energy of a photon is given by

E = h

= hE

Where is the frequency of the radiation

Using de Broglie’s relation the wavelength of the photon is given by

= ph

Where p = m v = momentum of the particle.

The same principle may be applied to electrons also.

Now equation (3) can be written as

=Et pxA exp 2 ih h

= iA exp Et px .... (4)

Where h = h/2

This expression for is correct only for freely moving particles. Differentiatingpartially with respect to x, we get

ipeAipx

)pxEt(i


(or)

pxi

Similarly,

22

22 p

x .... (5)

Now

Ei

t

(or)

ti

h = E .... (6)

The total energy of a particle is the sum of its kinetic energy and potential energy.

i.e. E = ½ mv² + V = m2

p2

+ V ....(7)

Substituting the values of p² from equation (5) and E from equation (6) inequation (7) we get

V

xm2–

ti 2

22

i.e.

V–

xm2ti 2

22 ....(8)

This can be written in three dimensions as

)t,z,y,x(V)t,z,y,x(zyxm2

)t,z,y,x(ti 2

2

2

2

2

22

(or)

V

m2ti 2

2 ....(9)

Where ² = 2

2

2

2

2

2

zyx

= Laplacian operator

Equations (8) and (9) are the famous time dependent Schroedinger waveequations in one dimension and three dimensions respectively.


Derivation of time independent Schrodinger wave equation

If we apply this equation for stationary state problems, in which the potential of aparticle does not depend upon time explicitly and the forces that act upon it and henceV, vary with position of the particle only, we get simplified time independentSchroedinger wave equation as follows.

Let us postulate a solution of the form

(x, t) = (x) (t)

i.e. (x, t) = iEtiEtipx

e)x(eeA

2

2

x

= e–iEt/ 2

2

dx)x(d

t

=

iE–

e–iEt/ (x)

Substituting these values in the one dimensional equation (8) we get

i h

iE–

e–iEt/ (x) = m2

– 2 e–iEt/ 2

2

dx)x(d

+ V (x) e–iEt/

E (x) – V (x) =m2

– 2 2

2

dx)x(d

(or) 2

2

dx)x(d + 2

m2 ( E – V ) (x) = 0 .... 10

In three dimensions, the above equation can be written as

² + 2m2 ( E – V ) = 0 .... 11

Where is a function of x, y and z only.


1.10 APPLICATIONS OF SCHROEDINGER WAVE EQUATION1. Energy levels of an electron in an infinitely deep potential well (in one

dimension):

Consider an electron which is placed in an infinitely deep potential well with finitewidth ‘a’. We assume that the movement of the electron is restricted by the sides of thewalls and the electron is moving only in the x direction; when it collides with the walls,there is no loss of energy of the electron and so the collisions are perfectly elastic(figure 1.6).

Since the electron is moving freely inside the well its potential energy V = 0. Butthe potential energy V of the electron is infinitely high on both sides of the well andoutside the well also. Due to that the electron cannot escape from the well through thesides.

V =

V = 0

x = 0 x = a

V =

Figure 1.6: An one dimensional potential wellwith walls of infinite height at x=0 and x=a

Boundary conditions

1. Since the potential outside the well is infinitely high, the probability of findingthe particle outside must be zero.

i.e. ||² = 0 0 > x > a

Therefore = 0 at x = 0 and x = a

2. Inside the well the wave function is finite

i.e. ||² 0 0 < x < a


The one dimensional Schroedinger wave equation is given by

22

2 m2dxd

(E – V) = 0

Here V = 0 and E is simply equal to kinetic energy of electron.

22

2 m2dxd

E = 0

(or) d² /dx² + k² = 0

where k² = 2

22

22

4m2

pm2mE2

(using de Broglie’s relation)

[k is called wave vector or wave number. | k | = 2/]

The above equation is similar to the equation of harmonic motion and so thesolution can be written as

= A sin k x + B cos k x ....(1)

To evaluate the constants A and B we must apply the boundary conditionsnamely = 0 at x = 0 and x = a

When x = 0, we get

= 0 = A sin(0) + B cos(0)

B = 0

When x = a,

= 0 = A sin k a

ka = n

(or) k = an

But k² = 2

mE2

2

mE2 = 2

22

an


or En = 2

222

ma2n

= 2

22

ma8hn

....(2)

and n = A sin axn

....(3)

Let us find the value of A.

Obviously a

o

|n |² dx = 1

Since the electron should exist within the well; substituting the value for n,we get

a

o

A² sin ²

axn

dx = 1

A² a

o

2anx2cos1

dx = 1

1anx2sin

n2ax

2A

a

o

2

i.e. 2

aA2

= 1

or A = a/2

n = a/2 sin

an

x ....(4)

According to equation (2) the energy values of the electron are discrete. So thatelectron will be in any one of the above energy states or eigen states n at a given time.These energy values are often referred to as eigen values or allowed values and occurin all quantum mechanical problems concerning spatially bound or constrained particles;the number n is called a quantum number. The lowest eigen state is called the groundstate.


Here the lowest energy level E1 = 2

2

ma8h

E

2

1

3

0 a1

E2

E3

E

x 0 ax

2| |2| |32| |

22| |

12| |

n2| |

Figure 1.7: The first three electron energy levels and their correspondingwave functions and probability densities are represented diagrammatically.

Results

1. The energy is quantised and so it cannot vary continuously.

2. For the same value of the quantum number ‘n’, the energy is inverselyproportional to the mass of the electron and to the square of the width of thewell. The permitted energies of an electron confined in a well 1 Å wide are

n² × (6.63 × 10-34)²En = joule

8 × 9.1 × 10-31 × 10-20

= 38 n² eV

E1 = 38 eV, E2 = 152 eV, E3 = 342 eV, etc.

These energy levels are evidently quite far apart to make the quantization ofelectron energy in such a well conspicious. If however the well is of macroscopicdimensions, say 1 cm wide, the permitted electron energies areEn = 38 × 10-16 n²eV. The permissible energy levels are now so close togethersuch that they may be appeared to vary in a continuous manner.

3. The probability of finding the electron in the first energy level is maximum atthecentre of the well. But in the second energy level it is zero at the centre ofthe well. Thus in each energy level, the location of finding the electron isdifferent (figure 1.7).


2. Electron in a metal

Consider the same problem for a three dimensional metal in which the electronsmove in all directions so that three quantum numbers nx, ny, and nz are needed,corresponding to the resolution of the motion into components along three perpendicularaxes x, y and z. For simplicity we will take the potential energy of the electron to bezero inside the metal and infinite outside. So that the problem is just an extension of theone dimensional problem discussed earlier. Therefore with a cubically shaped block ofmetal of sides ‘a’ the permitted energy levels can be written as

En = 2

2

ma8h

(nx² + ny² + nz²) .... (A)

where nx, ny and nz can each take any number from the set 1, 2, 3, . . . etc. irrespectiveof what numbers the others take.

Similarly the wave function

nx, ny, nz = 3a8

sin xn xa

sin yn ya

sin zn za

.... (B)

The three quantum numbers nx, ny and nz are required to specify completely eachstationary state. It should be noted that the energy E depends only on the sum of thesquares of nx, ny and nz.

Consequently there will be in general several different wave functions having thesame energy. For example the three independent stationary states having quantumnumbers (2, 1, 1), (1, 2, 1) and (1, 1, 2) for nx, ny and nz have the same energy value

2

2

ma8h6

. Such states and energy levels are said to be degenerate and the corresponding

wave functions are 211, 121 and 112.

On the other hand if there is only one wave function corresponding to a certainenergy, the state and the energy level are said to be non - degenerate. For example

the ground state with quantum numbers (1, 1, 1) has the energy 2

2

ma8h3

and no other

state has this energy. The degeneracy breaks down on applying a magnetic field orelectric field to the system.


1.11 FERMI ENERGYThe expresion for the energy values given in equation corresponds to the permissible

energy values that the valence electrons in a metal may have, but it is essential to knowwhat energies the electrons actually possess. For a piece of metal of macroscopicdimensions, say a centimetre cube, the energy of the ground state(nx = ny = nz = 1) is of the order of 10–15 eV and hence may be taken to be zero forall practical purposes. Also the maximum spacing between consecutive energy levels isless than 10-6 eV. So the distribution of energy levels may be regarded as a continuum.If a plot is made for a large range of energy values such that the individual En valuesare so close together that they can be shown as a continuum and the number of statesper interval of energy, N(E) increases parabolically with increasing E as shown infigure 1.8 (a). The dashed line shows the nature of the change in electron energies thatoccurs on heating to room temperature.

N(E)

0K

E EF

E

E F

At roomtemperature

Fermi level

(a) (b)

(a)The distribution of energy states as a function of energy E(b) Filling of energy levels by electrons at 0 K

Figure 1.8The valence electrons tend to occupy the lowest available energy states. However

because of the mutual interactions among all the electrons that form the electron gas, itis necessary to consider that all the electrons are in a single system and that the PauliExclusion Principle applies; according to which only two electrons can occupy a givenstate specified by the three quantum numbers (nx, ny and nz), one with spin up and otherwith spin down (i.e. with opposite spins). As a result of this principle, at 0 K the electronsfill all the states upto a certain maximum energy level, Emax called the fermi level orfermi energy ‘EF’. All quantum states in the energy levels above EF are empty(figure 1.8 (b). Thus the fermi level is a boundary line which separates all the filledstates and empty states at 0 K.


Thus the energy of the highest filled state at 0 K is called the fermi energy‘EF’ or fermi level. The magnitude of EF depends on how many free electrons thereare. AT 0 K all states upto EF are full and states above EF are empty. At highertemperatures the random thermal energy will empty a few states below EF by elevatinga few electrons to yet higher energy states. No transitions to states below EF will occursince they are full. Thus an electron cannot change its state unless enough energy isprovided to take it above EF. The probability p(E) of an electron occupying a givenenergy level is represented by

p(E) = )kT/)EE(( Fe11

– Fermi - Dirac distribution function .... 4.23

At 0 K, p(E) = 0 for E > EF

= 1 for E EF

Unit probability means that the state is always full, zero probability means that itis always empty and a fractional probability means that it is full for part of the time. Itcan be seen from figure 1.9 (a) that at 0 K the electrons move about with kineticenergies of all values upto EF. At temperatures above 0 K some electrons absorb thermalenergy and move into higher quantum states. The Pauli exclusion principle rules that anelectron can only enter an empty state, so that the thermally excited ones must go intostates above EF. Thermodynamics shows that the average allowance of thermal energyto a particle in a system at a temperature T K is of the order kT so that only electronswithin an energy interval kT from EF, approximately, can take up thermal energy and goto higher states with energy EF + kT (figure 1.9 (b)).

p(E)

0EFE

1

p(E)

0EFE

1.0

0.5

kTa. At 0 K b. At T K where T > 0

Figure 1.9: The fermi distribution curve

At room temperature kT is only 10-2 EF or less, so that only something like 1% ofthe electrons can take their allowance. That is why the specific heat of the electron gasis much smaller than what the classical free electron theory predicts.


1.12 EFFECTIVE MASSEffective mass of electrons in metals

Generally in most conductors m* = m since the band is only partially filled. Butthe effective mass of electron in metals like Copper, Magnisium and Platinum is greaterthan the mass of free electron. So if we substitute effective mass of electron instead oftrue free electron mass in the expressions for specific heat, electrical conductivity andthermal conductivity, we can get the correct values.

The concept of effective mass is able to account for many experimentalobservations like high electronic specific heat of transition metals and their highparamagnetic susceptibilities.

Effective mass of electron in semiconductors and insulators

The effective mass plays an important role in the conduction process insemiconductors and insulators since they have full or almost filled valence bands. Wecan find that the effective mass m* is negative near the zone edges of almost filledvalence bands. Physically speaking the electrons in these regions are accelerated in adirection opposite to the direction of the applied force. This is called the negative massbehaviour of electrons.

The electrons with negative mass can be considered as a new entity having thesame positive mass of that electron and the same positive charge as the numericalvalue of the electron’s charge. The new entity is given the name ‘hole’. The advantageof the concept of positive holes is that the momentum and current of a nearly filled bandwith n empty states can be attributed to the presence of an equivalent number of nholes with the same positive mass and positive charge of that of electron.

E

e

h

ev ej

hjhv

Figure 1.10: Motion of electron in the conduction bandand holes in the valence band in the electric field ‘E’


The holes are not real particles like electrons or positrons, but it is only a way oflooking at the negative mass behaviour electrons near the zone edge. We look upon themotion of the effective negative mass electrons as the motion of the positive holes orpositive vacant sites in a nearly full band and allow the electrons in the band to carry thecurrent.

The positive hole conduction and effective negative electron mass conductionare equivalent situations (figure 1.10). Calculations made on the hole picture isadvantageously retained. Several phenomena like Hall effect, Thomson effect, etc. findready explanation on the basis of the hole concept.

1.13 DENSITY OF STATESDensity of states ‘G(E)dE’ is defined as the number of states per unit volume in

an energy interval. E and E + dE.

No. of states between E and E + dE in a metal pieceG(E) dE =

Volume of that metal piece

zn

EE + dE

n

no

n

y

x

Figure 1.11: The positive octant of n - space

The number of states with a particular value of E depends on how manycombinations of the quantum numbers result in the same value of n.


Since we are dealing with almost a continuum of energy levels, we may constructa space of points represented by the values nx, ny and nz and let each point with integervalues of the coordinates represent an energy state. Let us calculate the density ofstates in a cubical metal piece with sides ‘a’. Let nx, ny and nz be the coordinate axes.

Draw a sphere in these axes with radius n² = nx² + ny² + nz² and energy E. Thissphere contains a series of shells. Each shell denotes a particular radius (or) particularenergy value. Any change in the nx, ny, and nz will change E and hence the radius ‘n’.Suppose we want to find the number of states in between E and E + dE.

This is equivalent to say that the number of states in a shell thickness n at adistance n in the coordinate system formed by nx, ny and nz. Since nx, ny and nz will takeonly +ve values, in that sphere 1/8 of its volume will satisfy this condition. Further inthat octant we require only the shell thickness of n at a distance n.

G = Number of states in the shell of thickness n at a distancen from origin

= 1/8 × 4 n² n

= 2n²n

We know that E = 2

22

ma8hn

Differentiating, we get

dE = 2n dn 2

2

ma8h

ndn = 2

2

h2ma8dE

From equation we get

n = h)ma8(E ½2½

G = 2

h

)ma8(E ½2½

2

2

h2ma8dE

= 2

3

2/3

h2)m8(

a³ E½ dE


Since volume of the metal V = a³,

G = 2/33 )m8(

h4

V E½ dE .... (1)

According to pauli’s exclusion principle, in each state 2 electrons can beaccommodated.

Therefore if all the states are filled up by electrons in the energy interval (betweenE and E + dE) then the number of electrons per unit volume in that interval

dN = g (E) dE = 2 3h4

(8m)3/2 E ½ dE

= 3h2

(8m)3/2 E ½ dE .... (2)

Normally all the states are not filled sates. The probability of occupation of electronin an energy state is given by Fermi-Dirac distribution function p(E).

dN = p(E) g (E) dE .... (3)Calculation of density of electrons at 0 KAt 0 K, p (E) = 1

dN = FE

03h2

(8m)3/2 E ½ dE

N = 3h2

2/3

E)m8( F2/3

N = 3h3

(8m) 3/2 EF3/2

=weight Atomic

number Avagadro density atomper electros free ofnumber

(or) EF =3/2

2/3

3

)m8(Nh3

=

m8h2

3/2N3

....(4)

Hence the Fermi energy of a metal depends only on the density of electrons ofthat metal.


Table 1.1: Electron density and Fermi energy of various metals

MetalElectron density ‘N’ Fermi energy ‘EF’ at 0 K

× 1028/m³ eV

Na 2.50 3.1

Ag 5.76 5.5

Cu 8.50 7.0

Zn 13.10 9.4

Al 18.10 11.6

The above table shows the values of electron density and Fermi energy of variousmetals calculated from the above equations. Here we have derived the Fermi energy at0 K. But when the temperature increases, the Fermi level or Fermi energy slightlydecreases.

It can be shown that

EF = EF0

2

F

2

0EkT

121

and EF0= kTF where EF0 is equal to Fermi energy at 0 K and TF is called

Fermi temperature.

1.14 ORGIN OF BAND GAP IN SOLIDSBand gap is the net gap considering all directions along which the electrons cannot

take those values of energy that lie in that gap regardless of their direction of motion.But this band gap disappears when there is sufficient overlap in the energy bands fordifferent directions.

Valence bandValence band

Conduction band

VE

CE

a. Partially filled valence band in conductors b. Overlapping bands in conductors

Figure 1.12: Schematic band structure of conductors


In a solid, outermost band that is fully or partially filled is called the valence band.The band that is above the valence band and that is empty at 0 K is called the conductionband. Solids can be classified on the basis of their band structure as conductors,semiconductors and insulators.

Conductors are those solids which have vacant electron energy states immediatelyabove the highest filled level of the valence band. This can happen in two ways. In thefirst case, the valance band is only partially filled as in figure 1.12(a). The electronshere can respond to an externally applied field by acquiring extra velocity and movinginto higher energy states.

In the second case, a full valance band overlaps the conduction band as shown infigure 1.12(b) so that the forbidden gap is zero. Monovalent metals such as the alkalimetals have one electron per atom in the outer most shell and the outer most energybands are half filled in these metals. Divalent metals such as magnisium have overlappingconduction and valence bands. Therefore they can also conduct even if the valenceband is full. The band structure of trivalent metals such as Aluminium is similar to thatof monovalent metals.

Valence band

a. Energy band of silicon (semi - conductor)

Conduction bandEg = 1.1 eV

VECE

Valence band

Conduction band

Eg = 5.4 eV

VE

CE

b. Energy band of diamond (insulator)

Figure 1.13: Band structures of semiconductor and insulator

Semiconductors are those materials which have an energy gap of about 2 to 3 eVor less. When the energy gap is 2eV or less an appreciable number of electrons can beexcited across the gap at room temperature. So semiconductors conduct much betterthan insulators at room temperature but still orders of magnitude poorer than metalswhich have no forbidden gap. By adding impurities or by thermal excitation we canincrease the electrical conductivity in semiconductors.


Insulators are those materials which have an energy gap more than 3eV. It hasbeen estimated that millions of volt/m of electrical potential would be necessary toaccelerate an electron sufficiently to jump the forbidden gap. The other possibility for atransition is that electrons cross the gap by thermal excitation. At room temperature thenumber of electrons that can be thermally excited across the gap in insulators such asDiamond turns out to be extremely small. So the conductors are the materials havingenormous electrical conduction; the insulators are the materials in which practicallythere is no electrical conduction and the semiconductors are the materials in which theelectrical conduction is in between the electrical conduction of conductor and insulator.

Energy Bands in Solids

Quantum free electron model gives the electrical conductivity of metals correctly.But it is unable to give the electrical conductivity of semiconductors and other propertiesof solids. Due to that the band theory of solids or zone theory of solids is developed.

According to that when an electron moving through a periodic lattice, its mass isnot only converted into its effective mass me *, but also the band gap arises. When wehave crowded number of electrons, due to overlapping of electrons, each energy levelis splitted into N energy levels and these splitted energy levels should be accommodatedwithin a small region. Thus the crowded splitted energy levels appear as a band.The band is formed only in solids.

The first allowed band is formed from K shell electrons and second allowed bandfrom L shell electrons. The valence electrons from the valence band which is the topmost filled band. Above the valence band there is conduction band which consists offree electrons or conduction electrons. At 0K, there is no free electron in it. As thetemperature increases or the applied field increases the number of electrons in conductionband will increases further.

In between the conduction band and the valence band there is forbidden gap orband gap. The energy levels available in this gap or not allowed to occupy by the electronsof that solid. When the band gap is more, there is little conduction. In metals there is noband gap. So there more electrical conduction. In semiconductors, the values of bandgap are in between 0.5 eV to 2 eV. In insulators the value of band gap is more than3 eV. The value of the band gap of semiconductor can be altered by adding or dopingsome impurity atoms belonging to III group and V group of elements in the periodictable.


1.15 1 D SCATTERING OF ELECTRONS IN PERIODIC POTENTIALWhile the Sommerfeld theory discussed earlier accounts satisfactorily for electrical

conductivity in most metals, it failed to explain why other substances that also containfree electrons have virtually no conductivity and are considered to be excellent insulators.A solution to this problem is given by the Zone Theory or Band theory of solids.

The effect of the periodic lattice field on the motion of the electrons leads to thezone or band theory of solids, which is of the greatest importance for understanding thestructures and properties of metals, alloys and non-metallic solids. According to zonetheory, the electrons move in a periodic field provided by the lattice.

The potential of the solid varies periodically with the periodicity of space latticeand the potential energy of the electron is zero near the nucleus of the +ve ion in thelattice and maximum when it is half way between the adjacent nuclei which are separatedby the interatomic spacing distance ‘a’.

This model was first postulated by Kronig and Penny. So taking this model andsolving the Schroedinger equation for this case, we can find the existence of energygap between the allowed values of energy of electron. So if we use classical theory, wecan get a parabola when we plot the curve between the electron’s energy and itsmomentum.

Since the curve is a parabola, we can infer that the energy varies continuously.But by Kronig - Penny model, we can get a parabola with some discontinuties in it asshown in figure 1.14(b).

E

o k

Forbidden band

Secondallowed band

Firstallowed band

B

A

EB

AEg

koa

a–

a. Classical free electronmodel energy curve

b. Kronig-Penny modelenergy curve

Figure 1.14: Energy Vs momentum curve


SOLVED PROBLEMS1. Electrons are accelerated through 344 volt and are reflected from a crystal.

The first reflection maximum occurs when the angle between it and thenormal to the crystal is 40o. Calculate the spacing between the principalplanes of the crystal. Assume the electrons are incident normal to thecrystal.

de Broglie wavelength V26.12

Å 34426.12

Å = 0.66 Å

Now sind2n

sin2nd

oo 401802 = 70o

Substituting the values of and , we get, o

1 0.66d 0.352 sin 70

Å

2. Compute the de Broglie wavelength of 10 keV neutron. Mass of one neutronmay be taken as 1.675 × 1027 kg.

Kinetic energy of neutron= 10 keV = 104 × 1.6 × 10-19 joule= 1.6 × 10-15 joule

1/2 mv2 = 1.6 × 10-15

v = s/m1038.110675.1106.12 6

2/

27

15

00286.0m1086.21038.110675.1

10625.6mvh 13

627

34

Å

3. Compute the energy difference between the first and second quantum statesfor a free electron in a solid 1 metre cube.

E = 2 2 2

x y z22

n n nh

8 ma


For the first quantum state nx = ny = nz = 1. Also a = 1 m

E1 = 2–34 2 2 2

–31 2

6.63 10 1 1 18 9.1 10 1

= 1.81 × 10–37 joule

There are many equal energy states above the first state having nx, ny and nz as(1, 1, 2) (1, 2, 1) and (2, 1, 1).

For all these states nx2 + ny

2 + nz2 = 6

E2 = 2–34

–31 2

6.63 10 68 9.1 10 1

= 3.62 × 10–37 joule

E2 – E1 = 1.81 × 10–37 J

4. An electron and a bullet are independently moving with 300 metre/second,accurate to 0.01%. The bullet mass is 0.05 kg. With what fundamentalaccuracy can we locate their positions? Comment on your results.

For electron:

Momentum of electron p = mv = 9.1 × 10-31 × 300 = 2.7 × 10-28 kg-m/s.

The uncertainty in momentum is given to be 0.01%. Here the mass of the electronis taken to be a constant.

(or) p = 0.0001 × 2.7 × 10-28 = 2.7 × 10-33 kg

The minimum uncertainty in position

x = m002.04107.2

106.64p

h32

34

.

If we consider the electron as a dot, then there is no meaning for the determinationof position of the electron with accuracy about 0.2 cm. Other wise when wemeasure the momentum with accuracy about 0.01%, then we cannot locate itsposition accurately.


For Bullet:

Momentum of the bullet, p = mv = 0.05 × 300 = 15 kg-m/s

and p = 0.0001 × 15 = 1.5 × 10-3

x = metre104.04105.1

106.6 313

34

.

The accuracy in determining the position of the bullet is very high. This is so farbeyond the possibility of measurement and we can assert that for heavy objectslike bullets, the uncertainly principle sets no limit whatever on our measuringtechniques. Therefore the uncertainty principle is applicable to microscopic bodieslike fundamental particles.

5. A particle is moving in a one-dimensional box (of infinite height) of width10 Å. Calculate the probability of finding the particle within aninterval of 1 Å at the centre of the box, when it is in its state of leastenergyThe wave function of the particle in the ground state (n = 1) is

Lxsin

L2

1

The probability of finding the particle in unit interval at the centre of the box(x = L/2) is given by

22

1 L)2/L(sin

L2

= L2

2sin

L2 2

The probability of finding the particle within an interval of x at the centre ofthe box

= x)L/2(x21

= 2.01010102

10

10

.


6. Calculate the de Broglie wavelength of an electron accelerated to apotential of 2 kV.

Relativistic variation of mass is not significant at 2 kV.

Kinetic energy = 1/2 mv2 = 2 kV = 2000 × 1.6 × 10-19 J

Mass of electron = 9.1 × 10-31 kg

Momentum = p = mE2

= 1931 106.12000101.92

de Broglie wavelength = h / mv = h / p

= 1931

34

106.12000101.92

106.6

= m1073.2 11

7. Calculate the wavelength associated with a thermal neutron of energy0.025 eV

Mass of neutron M = 1.676 × 10-27 kg. 1 eV = 1.602 × 10-19 J

1/2 mv2 = 0.025 × (1.602 × 10-19)

or mv = [2 × 0.025 × 1.602 × 10-19 M]1/2

2/12719

34

)]10676.1()10602.1()025.0(2[1062.6

mvh

= 1.807 × 1010

m = 1.807 Å

Documents

Quantum Mechanics - drmgriyearbtech.com · 2020. 2. 10. · Davisson - Germer Experiment Experimental arrangement In 1927, Davisson and Germer for the first time proved the wave nature