quantum mechanics basics

8/12/2019 Quantum Mechanics Basics

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ince wave nature of the particle are confirmed we may also choose simpleform of wave to represent the matter wave

x , t=sin2 x t!e are familiar with such waves in a string under "artesian coordinates

#f we consider Bohr model of atoms such wave exist in sphericalcoordinates$ therefore we may choose generalized coordinates for theformalism of the wave-mechanics with "artesian as special case%

chroedinger equation may be regarded an equation that gives form of thewave under correct potential% &or example in Bohr model it is sphericallysymmetrical potential


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chroedinger equation is and equation from first principles of experimentalobservations listed as andE=h =

h

p

#t should be consistent with relation with energy momentum of a classicalsystem

E= p

2

2mV

ince quantum waves interfere same as in the way classical wave thesolutions they should be linear as in classical waves such as water waves%

Therefore if and are two independent solutionof the differential equation then

also must be solution to this equation ee Davisson and (ermerexperiment for interference of quantum waves

1x , t 2x , t

x , t=C11x , tC22x , t


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x , t=sin2 x t

x , t=sin k x t

)sing the change of variable k=2

=2

2 k2

2mVx , t=

#n this equation if expressed in the form of the differential equation inorder to get in the *+ we need second order differentialoperator with space

k2

,lso in order to get on + we need a differential equation that

is first order in time%

!e may propose an empirical form of the equation of a matter wave

2x , t

x2

Vx , t x , t= x , t

t


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2x , t

x2 Vx , t x , t=

x , t t

!here and are undetermined constants

x , t=sin k x t

!ith the trial solution under constant potential the differentialequation becomes

sin k x tk2sin k x tV0=cos k x t

V0

This is possible in the special case of where

cos k x t=sin k x t

x , t

This difficulty can be overcome by using a trial solution that is acombination of sine and cosine waves

x , t=cos k x t sin k x t


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ubstituting in the equation with constant potential

2x , t

x2

Vx , t x , t= x , t

t

k2cos k x t k2 sin k x t

V0cos k x tV0 sin k x t=

sin k x tcos k x t

k2V0 cos k x t

.lacing together the coefficients of sine and cosine functions

k2 V0 sin k x t=0

/eaningful solutions are possible when the coefficients vanishleading to two algebraic equations

2


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k2V0 cos k x t

k2 V0 sin k x t=0

k2V0=

The two equations are

k2V0=/

By subtracting one from other

0=/

=1/

2

=1

=i

)sing this result we may obtain value of other constants


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k2V0= =i

k2V0=i

By comparing with the equation equating different parts of the energy

2 k2

2mV0=

=2

2m

=i

=i

0o specific difference is there for selection of either positive ornegative sign for beta$ therefore choosing positive sign as choice

=i


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Meaning and properties of wave functions

!ave functions give state of a system$ the time evolution of a classicalstate is given time variation of the wave functions

is interpreted as the probability amplitude of the particle'spresence

r , t

.osition vector in a three dimensional space

!ave function contains all information about the state of system

r , t

r can ta1e any value in the space

d P r , t .robability of finding a particles in thein the volume element d r

3=dx dy dzr , t

2

#s interpreted as probability density


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d P r , t=Cr , t2

d3

r

#s interpreted as probability density

0ormalization constant

r , t

"onsider the wave function representing total solution of the chrodingerequation

r , t=c11 r , tc22 r , t.......cn r , t

The total solution of the system may be decomposed in terms multiplestates in which system exist

r , t=i=1

n

c i i r , t

complex coefficient


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Ze

e

r

v n=1 n=2

n=3

#n the case of atomic orbitalswe may identify the following

1r , t

2r , t

3r , t

r , t=i=1

n

c i i r , t

0ormalized wave function

.robability finding particle in a

specific i th orbital is given by

Pi r , t= c i

2

i=1

n

c

i

2


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2

2m2 r , tVr , tr , t=i

r , t

t

The general form the chrodinger equation for three dimensionalspace

This is a fully phenomenological equation that can be arrived only byargument validity of which can verified only by experiments%

&or a system consists of one particle the total probability of finding a

particle anywhere in space is given by

d P r , t=1#n this case the wave function must be square integrable$ that is

r , t

2d

3r is finite

r , t2 d3 r= 1C

isnormalization constant

r , t2

d

3

r=1 for normalized wave

function


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The state of classical particle at time is determined by positionand the momentum

t rp

#n 2uantum mechanics infinite number of parmeters is associated withthe state of a particle at any time 3infinite number of pointsare in this function4

r , t

Quantum mechanical description of the free particle

chroedinger equation for a free particles

2

2m2 r , t=i r , t

t

This differential equation is satisfied by the equation of the form


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This differential equation is satisfied by the equation of the form

2

2m

2r , t=i

r , t

t

,ccording to de Broglie hypothesis a plane wave should satisfy thecondition

=2k2

2mE=

p2

2m!hich is equivalent to classical condition

This differential equation can have trial solution

r , t=A ei kr t

r , tr , t=A eikr tA eikr t2

r , t2=A

2

.lane wave predict uniform probability in the space

wavevector

i i l f iti th t l ith l th i


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principle of super positionsays that plane wave with any wave length isa solution to chroedinger equation

r , t=A11r , tA22r , t...An nr , t ............

An=An k

0ormalization factor of the wave-function at a particular momentum

&or free particle there must be a continuous distribution of wavelengths$then the summation can be replaced with an integral

x , t= 1

21/2 g k ei k x kt d k

, constant arising from the normalizing factor of &ourier transform 5d

0ow for the sa1e of simplicity let us assume plane waves parallel toaxis

x

r , t= 1

23/2 g keikr kt d3 k

#f we choose at a particular origin of time this wave can be further


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#f we choose at a particular origin of time this wave can be furthersimplifies as

x ,0= 1

2

1/2 g k ei k x

d k

This is the mathematical representation a &ourier transform of function$therefore the validity of the formula is not limited to free particle%

g k

Form of wave packet at given time

k

k0 k

gk

Behavior of such wavepac1ets may be studies using

simple case of interference ofwaves


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The qualitative behavior of the interference of the wave can be identifiedwith interference of three waves

*et the wavenumber corresponding these three waves are given by

k0 , k0 k2

k0 k2

,

, nd let their amplitudes are proportional respectively12

,1, 12

!e then have

x , t= g k021/2

ei k0x1

2e

i

k0

k

2

x

12

ei

k0

k

2

x

=

gk0

2

1/2e

i k0x

1cos k2 x


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The interference between completely destructive when the phase


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The interference between completely destructive when the phasedifference between

ei k0x

ei k0 k2xand

out of phase

e

i k2 x=e

i

k2x=That mean between two destructive interference we can haverelation for the products

kx=4

, single particle moving in space given as interference of collection waves

of different wavelengths and frequencies which constructively interfere in aregion around mean position of the particle from this relation we mayderive a relation between mean change in position and momentum of theparticle

kx=4


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kk0xx0=4

mean wavenumbermean position

2

20

xx0=4

/ultiplying both sides with .lanc1's constant

2 h

2 h0

xx0=4 h

pp0xx0=2h

p x=2h

This means that when we measure the momentum or position of aparticle simultaneously we cannot measure the with accuracy

specified by this relation


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This result is spacial case of the +eisenberg )ncertainty principlewhich is in the form of an inequality

p x

2Ta1e the case of a single plane wave

x ,0=A ei k x

ince this wave have a well defined wavenumber the the uncertaintyin the position is infinite

#n the similar fashion for a point particle the uncertainty in themomentum is infinite

#n stead of considering a stationary wave$ if we ta1e the case of timevariation only in a wave we can get an uncertainty relation connectingenergy and time associated with wave

E t

2

Uncertainty relation and atomic parameters


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y p

Vr = 1

40

e

2

r

"onsider an electric field in the "oulomb field of an electron

Vr =k2

r

,ll constants of the problem

,ssume that the wavefunction of ground state is described by a sphericallysymmetrical wave function let the average position of the electron is given by

r0

Therfore the potential energy becomes Vk2

r00ote that as electrons have wave nature position is not exact

#f electron is confined in a space having dimension the momentum isgiven by

r0

p x2

#f electron is confined in a space having dimension the momentum isi b

r0


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given by

p x2

p r0The average momentum in this case is zero however average 1inetic energyis not zero

TTmin=p

2

2m =

2

2m r02

Total energy compatible with the uncertainty relation is given by

E=T V= 2

2m r02 k

2

r0

Vk2

r0

The value of is given by minimum of this functionr0*et

fr =

a

r2

b

r0

b


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fr =a

r02

b

r0

d fr0

d r0=

2a

r03 br0

2

0=2a

r03

b

r02

/inimum is at value

b r0=2a k2

r0=2 2

2m

r0= 2

m k2

ubstituting this in energy relation

E= 2

2m r02

k

2

r0=

2

2m

2

m k2

2

k2

2

m k

2

E= 2 k2


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E=

2m 2

m k2

2

2

m k2

=m k4

22=

m k4

22

m k4

2

En=

1

4 0 2

m Z2

e4

22

1

n

2

Z=1,n=1

k= e2

40

E=

m

e

2

40

4

22

=

m

e

2

40

4

22

E=

1

4 0 2

m e4

22

This is the same expression we get the Bohr model for first orbit of+ydrogen


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#n quantum mechanics state of system is represented in +ilbert space it is a


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#n quantum mechanics state of system is represented in +ilbert space it is aform of the abstract linear vector space

r , t=i=1

n

c i i r , t

#n quantum mechanics state of system is represented in +ilbert space$each vector

, particular state of the system is given by i r , t

#n +ilbert space the inner product is defined as

i r , t Each vector is orthogonal toeach other

i

r , tj r , t d3

r

!hen the state vectors are orthogonal

ir , tj r , t d3 r=Cij


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#n the same manner in which we obtain the value of momentum we canobtain other observables li1e position and energy from the wave


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obtain other observables li1e position and energy from the wavefunction- ta1e case of 7d plane wave wavefunction

!x x , t=x x , t

!x=x

!x A ei kxx t=x A e ikxx t=x x , t

form of the operator

!E x , t=e x , t

The energy operator is given as

form of the operator !E=i t

i t

x , t=i t

A eik x t= A e i k x t

x , t=A ei k x t

Eigenvalue of energy

0ow ta1e the case of full wavefunction representing the state of a system


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,ccording theprinciple of super positionthe system exist in a superposition of all possible states of the system

r , t=

i=1

n

ci

i

r , t

n can have finite or infinite number of states

!hen the states are orthogonal we have the relation

ir , tj r , t d3 r=Cij

!hen they are orthogonal and normalized

ir , tj r , td3 r=ij!hen an observation is made on the system the system falls into one ofthe possible states of the system

r , t=i r , t

with probability ci2

0ow the postulate regarding the measurement can be stated as


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54 ,ny possible result of measurement of a physical quantity is oneof the eigenvalue of corresponding observable

!AA

#n the case of discrete solution of the wave function we can write

p g g

r , t=i=1

n

c i i r , t r , tr , t d3 r=1

!A i r , t=ai i r , t

Each of the state vectors obey eigenvalue equation

!hen the operator operates on the total wave function of the system the

probability finding a particular eigenvalue is given byP aiP ai=ci

2= ir , tr , td3 r

2

=

ir , t

i=1

n

ci i r , td3

r

2

:4 !hen the physical quantity is measured on a system in theA


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4 p y q y ynormalized state the probability of finding the non-degenerateeigenvalue of the corresponding observable is

where is the normalized eigenvector of associatedeigenvalue

r , tai

P ai= i

r , tr , td3

r2

i r , t !Aai

;4 time evolution of the state vector is governed by thechroedinger equation

r , t

!Hr , t=i

r , t

t

+amiltonian operator of the system< corresponding observable give total energy ofthe system

"#pectation value


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p

# r$= r P r , t d3 r

The expectation value is the mathematical expectation$ for the result of asingle measurement% 9n otherwise$ it is the average of results of large

number of measurements on an independent system

=

r , tr r , td3

r

Particle in a time$independent scalar potential


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eparation of variables in chrodinger equation

2

2m2

r , tVr , tr , t=i r , t

t*et us loo1 if there exist solutions of the form

r , t=%r &t

*et us loo1 if there exist solutions of the form

&t 2

2m2%r &tVr , t%r =i %r

&t

t

Dividing all terms by the relation %r &t

1

%r 2

2m2%r Vr , t=i

1

&t

&t

t

space dependent part time dependent part

1

% 2

2

2

%r Vr , t=i 1

&t


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The space dependent part and time dependent part of the can beequated if and only if both sides of the equation can be equated to aconstant we choose that constant to be

%r 2m % ,

&t t

1

%r 2

2m

2

%r Vr , t=i 1

&t

&t

t

1

%r

2

2m

2

%r Vr , t=

i 1&t

d&td t

=

The time dependent part can be solved get solution in the differentialequation as

i 1

d&t=


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i&t d t

' d&t&t =idt

&t=A ei tBy integration

!e may set the constant as one and move that part to space dependentsolution$ that is

r , t=%r ei t

imilarly the space dependent part should satisfy the differential equation

1

%r

2

2m

2

%r Vr , t=

2

2m

2

%r Vr , t%r =%r

, stationary state is state with well defined energy

2

=E


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2

2m

2

%r Vr , t%r =E%r

(2

2m2Vr , t )%r =E%r

#n the compact form

!H%r =E %r The *+ of the equation may be written as

9perator form of the +amiltonian

!H is a linear operator since!H1%1r 1%1r =1 !H%1r 1 !H%1r

!H%r =E %r Eigenvalue equation of a linear operator

%equired properties of eigenfunctions &wavefunctions'


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This is a fundamental requirement asthe evaluation of position andmomentum involves these conditionsfor conservation and for ma1ing thesequantities well defined

x finite

single valued

continuous

dx

d x

quantum mechanics basics

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