be phys 2010 11 quantum mechanics

8/2/2019 BE Phys 2010 11 Quantum Mechanics

1/62

QUANTUM MECHANICS

1BE-PHYSICS-QUANTUM MECHANICS-2010-11MIT- MANIPAL

TOPICS

Text BookPHYSICS for Scientists and Engineerswith Modern Physics (6th ed)

By Serway & Jewett

An Interpretation of Quantum Mechanics

One- Dimensional Wave Functions & ExpectationValues

The Particle Under Boundary Conditions

The Schrodinger Equation

The particle in a BOX A particle in a Well of finite Height

Tunneling Through a Potential Energy Barrier

The Scanning Tunneling Microscope

The Simple Harmonic Oscillator


2/62


Quantum Mechanics


3/62

Quantum Mechanics


Quantum Mechanics deals with WaveNature ofmicroscopic particles

It enables us to understand many

phenomena involving atoms, molecules,

nuclei, and solids


4/62

AN INTERPRETATION OF QUANTUM MECHANICS

Making a conceptual connection between particles and

waves.


For an electromagnetic radiation, we have the probabilityper unit volume of finding a photon in a given region ofspace at an instant of time as

Electromagnetic Wave Photon


5/62

Taking the analogy between electromagnetic radiation andmatter we can say

Theprobabilityper unit volume of finding theparticle

is proportional to the square of the amplitude of

the wave representing theparticle




6/62

The amplitude of the de Broglie wave associated

with a particle is calledprobability amplitude, or the

wave function, and is denoted by.6BE-PHYSICS-QUANTUM MECHANICS-2010-11MIT- MANIPAL


A wave function


7/62

In general, the complete wave function for a system

depends on thepositionsof all the particles in the system

and on timeand can be written as

(rj,t) = (rj) eitwhere rj is the position vector of the j

TH particle in the

system.




8/62

can be real or complex. But ||2 is always real and positive, and is proportional

to the probability per unit volume, of finding the particle

at a given point at some instant.

Ifrepresents a single particle, then ||2 (called theprobability density) is the relative probability per unit

volume that the particle will be found at any given point

in the volume.




9/62

The total probability of a particlebeing in the interval a x b is thearea under the probability densitycurve from a to b.



One-Dimensional Wave Functions

P(x) dx = ||2 dx is theprobability to find the particle inthe infinitesimal interval dxaround the point x

dxP2

b

a

ab


10/62

Normalization

Because the particle must besomewhere along the x axis, the sum ofthe probabilities over all values of x(from

to

) must be 1.

1dx

-

2



Forcing this condition on the wave

function is called normalization


11/62

Expectation Value



The average position at which one

expects to find the particle after many

measurements is called theexpectation valueof xand is

defined by the equation

dxxx

where * is the complex conjugate of


12/62

Expectation Value



The expectation value of any function

f(x) associated with the particle is

dx)x(f)x(f


13/62

The important mathematical features of a physically reasonablewave function (x) for a system are

(i) (x), must be finite, continuous and single valued everywhere;

(ii) The space derivatives of, must be finite, continuous andsingle valued every where;

(iv) must be normalizable.




14/62


SJ: P-SE 41.1 A wave Function for a particle

A particle wave function is given by the equation (x) = A eax2(A) What is the value of A if this wave function is normalized?

(B) What is the expectation value of x for this particle?


1dx

-

2


15/62


SJ: Section 41.1 P -1 A free electron has a

wave function where x is

in meters. Find (a) its de Broglie

wavelength, (b) its momentum, and (c) its

kinetic energy in electron volts.

)x100.5(i 10eA)x(



16/62

The Schrdinger Wave Equation



17/62

The Schrdinger Equation


Just like the electromagnetic waves,the waves associated with materialparticles also satisfy a waveequation.

The wave equation for material

particles is different from thatassociated with photons becausematerial particles have a nonzerorestenergy.

The appropriate wave equation formatter waves was developed byErwin Schrdinger.

Erwin Schrdinger


18/62

THE SCHRDINGER EQUATION

Schrdinger equation as it applies to a particle of mass mconfined to moving along x axis and interacting with its

environment through a potential energy function U(x) is


Where E is a constant equal to the total energy ofthe system (the particle and its environment).

= h/2

+

The above equation is referred to as the one-dimensional,time-independent Schrdinger equation.


19/62

Application of Schrdinger equation.

http://../I%20sem.%20BE-2006/quantum%20mechanics%20figs/class6&7/PIB.ppthttp://../I%20sem.%20BE-2006/quantum%20mechanics%20figs/class6&7/PIB.ppthttp://../I%20sem.%20BE-2006/quantum%20mechanics%20figs/class6&7/PIB.ppthttp://../I%20sem.%20BE-2006/quantum%20mechanics%20figs/class6&7/PIB.ppt


20/62

PARTICLE IN A BOX OR INFINITE SQUARE WELL

In figure (a), a particle ofmass mand velocity v, confined to

bouncing between two impenetrable walls separated by a distance

L is shown. Figure (b) shows the potential energy function for the

same system.



21/62



Because the walls are impenetrable, there is zeroprobability of finding the particle outside the box

Therefore,

() < >

() must be continuous in space. There must be nodiscontinuous jumps in the value of the wave function atany point. Therefore, if is zero outside the walls, it must

also be zero atthe wallsTherefore, ( )

Boundary Conditions on the wave function.


22/62



As long as the particle is inside the box, the potential energyof the system does not depend on the location of the particleand we can choose its constant value to be zero

Hence < < Outside the box, we must ensure that the wave function is

zero. We can do so by defining the systems potential energyas infinitely large if the particle were outside the box.

The only way a particle could be outside the box is if the systemhas an infinite amount of energy, which is impossible

Boundary Conditions on the potential energy.


23/62

Only those wave functions that satisfy these boundary

conditions are allowed.In the region 0 < x < L, where U = 0, the Schrdingerequation takes the form



+

< >

( )

< <


24/62

The most general form of the solution to the above equation is () + ()

where A and B are constants determined by the boundary andnormalization conditions.

Applying the first boundary conditions,

i.e., at , leads to()+() ,and at , , () + () + ,



,


25/62

since A 0 , sin(kL) = 0 .

; ( 1, 2, 3, . . )

Each value of the integer n corresponds to a quantizedenergy value, En, where



,

,,

C O O SQ


26/62

Energy level diagram for a particle

confined to a one-dimensional boxof length L. The lowest allowedenergy is

E1 = h2/8mL2.

According to quantummechanics, the particle can

never be at rest !!!.

This is the ground state energy for the particle in a box. Excited statescorresponds to n = 2, 3, 4 ---- have energies given by 4E1 , 9E1 , 16E1 ---.



,



27/62

To find the constant A, apply normalization condition


The corresponding wavefunctions are


( + )

, ,


28/62

Fig. (a) The wave functions for n=1, 2, and 3.Fig. (b) The probability densities for n=1, 2, and 3.



The first three allowed states for a particle confined to a one-dimensional box are shown next.


29/62

PARTICLE IN A BOX

SJ: Section 41.2 P- 10

A proton is confined to move in a one-dimensional box of

length 0.20 nm. (a) Find the lowest possible energy of the

proton. (b) What is the lowest possible energy for an electron

confined to the same box? (c) Account for the great difference

in results for (a) and (b)

Mass of proton: 1.67 X 10-27kg

Mass of electron: 9.11 X 10-31 kg



30/62

PARTICLE IN A BOX

See the sample problems P-SE 41.2, 41.3 and 41.4



31/62


A particle in a well of finite height

A PARTICLE IN A WELL OF FINITE HEIGHT


32/62


Potential energy diagram of a

well offinite height U and

length L. A particle is trapped in

the well. The total energy E of

the particle-well system is lessthan U


, < < , < >



33/62

The Particle energy E < U ;

Classically the particle is permanently bound in the potentialwell and cannot come out.

However, according to quantum mechanics, a finite probabilityexists that the particle can be found outside the well even if the

total energy E < U.

That is, the wave function is generally nonzero in regions I andIII. This is possible due to the uncertainty principle.





34/62

The Schrdinger equation outside the finite well in regions I and III is:

General solution of the above equation is



()

+



35/62

Solutionshould be finite.A must be 0 in Region III and B must be zero in Region I,otherwise, the probabilities would be infinite in those

regions.

Therefore, the wave functions outside the finite potentialwell are



< >

Thus the wave function outside the potential well decay

exponentially with distance.



36/62

Schrodinger equation inside the square well potential inregion II, where U = 0 is

General solution of the above equation is



() + ()



37/62

To determine the constants A, B, F, G, & the allowed valuesof energy E, we can apply the four boundary conditions and

the normalization condition.



() ()

=

= &

=

=

But the boundary conditions nolonger require that the wavefunction must be zero at the ends of the well



38/62

0x0x dxdx

d

)0()0(

III

III

d

LxLx dx

d

dx

d

)L()L(

IIIII

IIIII

Boundary conditions Boundary conditions





39/62




40/62

Tunneling through a Potential Energy Barrier


li h h i l i


41/62

Consider a particle of energy E approaching a potential barrier

of height U, (E < U). Potential energy has a constant value of U

in the region of width L and is zero in all other regions. This is

called a square barrier and U is called the barrier height.





42/62

Since E < U, classically the regions II and III shown in the figure are

forbidden to the particle incident from left. But according toquantum mechanics, all regions are accessible to the particle,regardless of its energy. The particle can violate classical physics by for a short time, ~ / .


u e g t oug a ote t a e gy a e



43/62

The wave function is sinusoidal in regions I and III but exponentiallydecaying in region II.The probability of locating the particle beyond

the barrier in region III is nonzero. The movement of the particle to

the far side of the barrier is called tunneling or barrier penetration.





44/62

The probability of tunneling can be described with atransmission coefficient T and a reflection coefficient R.

The transmission coefficient represents the probability that theparticle penetrates to the other side of the barrier, andreflection coefficient is the probability that the particle isreflected by the barrier.

Because the particles must be either reflected or transmittedwe have, R + T = 1.

An approximate expression for the transmission coefficient,when T


45/62


Applications of Tunneling Phenomenon

THE SCANNING TUNNELING MICROSCOPE


46/62

THE SCANNING TUNNELING MICROSCOPE


The scanning tunneling microscope (STM)enables scientists to obtain highly detailed

images of surfaces at resolutions comparable to

the size of a single atom.

An electrically conducting probe with a very sharp

tip is brought near the surface to be studied.

The empty space between tip and surface

represents the barrier, and the tip and surfaceare the two walls of the potential well.

Because electrons obey quantum rules ratherthan Newtonian rules, they can tunnel across

the barrier of empty space. If a voltage is appliedbetween surface and tip, electrons in the atoms of

the surface material can tunnel preferentially from

surface to tip to produce a tunneling current. In this way, the tip samples the distribution of

electrons immediately above the surface


47/62

SJ: P-SE 41.6 A 30-eV electron is incident on a square barrier ofheight 40 eV. What is the probability that the electron will tunnel

through the barrier if its width is (A) 1.0 nm? (B) 0.10 nm?





48/62

SJ: Section 41.6 P- 27

An electron with kinetic energy E =

5.0 eV is incident on a barrier with

thickness L = 0.20 nm and height U =

10.0 eV as shown in the figure.

What is the probability that the electron (a) will tunnel through the

barrier? (b) will be reflected?


g g gy


49/62

The Simple Harmonic Oscillator...



50/62

THE SIMPLE HARMONIC OSCILLATOR


The oscillatory motion of an object is said to be simpleharmonic, if the force acting on it is proportional to itsdisplacement from an equilibrium position and is actingin a direction opposite to the displacement of theobject

,


51/62





52/62

The potential energy of the system is,

Where the angular frequency of vibration is

/




53/62

Classical Results:

Classically, the particle oscillates between the points , where A is the amplitude of themotion.

In the classical model, any value of E is allowed, including E = 0,which is the total energy when the particle is in rest at x= 0.

Its total energy

+

( )




54/62

The solution of the above equation is given by

The constant B can be determined from the normalization

condition.


The Schrdinger equation for this problem is

+

/ and E=

/



55/62

The energy levels of a harmonic oscillator are quantized. The energy

for an arbitrary quantum number n is given by

The state n = 0 corresponds to the ground state, whose energy is

(); the state n = 1 corresponds to the first excitedstate, whose energy is E1 = (3/2), and so on.


+ + /



56/62

Energy level diagram for a

simple harmonic oscillator,superimposed on the

potential energy function.

The levels are equally

spaced with .The ground state energy is

().




57/62

The classical and quantum mechanical probabilities


The dotted curves represent the classical probabilities and the blue ones the

quantum probabilities for a simple harmonic oscillator.



58/62

The classical and quantum mechanical probabilities


The dotted curves represent the classical probabilities and the blue ones the

quantum probabilities for a simple harmonic oscillator.



59/62

In classical physics, probability densities are the greatest near the

endpoints of its motion where it has the least kinetic energy. This is

in sharp contrast to the quantum case for small n. In the limit of large

n, the probabilities start to resemble each other more closely as

shown in figure.

Planks equation for the energy levels of the oscillators differs from

the equation given by quantum harmonic oscillator only in the term

added to n. However this additional term does not affect theenergy emitted in a transition.


QUESTIONS QUANTUM MECHANICS


60/62

MIT-MANIPALBE-PHYSICS-QUANTUMMECHANICS-2010-2011 65

[MARKS]1. What is a wave function ? What is its physical

interpretation ? [2]2. What are the mathematical features of a wavefunction? [2]

3. By solving the schrdinger equation, obtain the wave-

functions for a particle of mass m in a one-dimensional box of length L. [5]

4. Apply the schrodinger equation to a particle in aone-dimensional box of length L and obtain the

energy values of the particle. [5]

5. Sketch the lowest three energy states, wave-functions, probability densities for the particle in a

one-dimensional box. [3]



61/62

MIT-MANIPALBE-PHYSICS-QUANTUMMECHANICS-2010-2011 66

[MARKS]

6. The wave-function for a particle confined to moving

in a one-dimensional box is

Use the normalization conditionon to show that [2]

7. The wave-function of an electron is

Obtain an expression for the probability of findingthe electron between x = a and x = b. [3]



62/62

[MARKS]

8. Sketch the potential-well diagram of finite height U

and length L, obtain the general solution of theschrodinger equation for a particle of mass m in it.[5]

9. Sketch the lowest three energy states, wave-functions, probability densities for the particle in apotential well of finite height. [3]

10.Give a brief account of tunneling of a particlethrough a potential energy barrier. [4]

11. Give a brief account of the quantum treatment of asimple harmonic oscillator. [5]

be phys 2010 11 quantum mechanics

Documents