genchem_lec03

BITS Pilani, Pilani Campus

CHEM F111 : General Chemistry

Lecture 3

BITS Pilani Pilani Campus

1

Summary (Lecture 2)

2

Foundation of Quantum Mechanics Why Quantum Mechanics ?

Wavefunction

Characteristics of an acceptable wavefunction

Single-valued

Continuous & differential

Bounded

Normalized

Observables & Operators


Probability Density is more Informative


Use of Eigen value Function

4

For determining the value of an observable (property, e.g. p, E)

for a physical system:

-Operate the operator (p, E) on well-behaved wavefunction (ψ)

of that system.

-This operator-function combination should form eigen value

equation and the corresponding eigen value = value of that

observable.

Information about the system could only be known if we know the

eigen value of observable.

A given wavefunction is an eigen function of some operators by not

an eigen function of others.

Not all observables are determined by any one wavefunction.


Energy: Hamiltonian Operator

5

E (total Energy) = KE + PE = mv2/2 + V = p2/2m + V

Vm

zyxVzyxm

xVdx

d

mV

m

p

22

2

2

2

2

2

22

2

222

2

),,(2

)(22

Laplacian operator

………..In 1-D

………..In 3-D Cartesian coordinates

Energy Operator : Hamiltonian ( H ) operator

p x = H


Time-independent Schrödinger Eq.

6

• Energy of a system could be known by operating total energy

operator on the wavefunction.

),,(),,(),,(2 2

2

2

2

2

22

zyxEzyxzyxVzyxm

Ĥ = E

• The wavefunction (for stationary states; independent of time)

satisfies the Schrödinger equation:

When is operated upon by Ĥ, returns the multiplied by E; a

category of equation called eigen (=own) value eqn.

Second order differential equation: the solutions will provide the

energies of possible states


7

General form of Schrödinger Equation

Observables & Operators

Acceptable wavefunction

Need for Quantum Mechanics

Next What ?


? How to extract information from the solutions

? How to define wavefunction of any system ?

? How to solve schrödinger equation for the system

Application to a Microscopic system

8

Translation Motion in 1-Dimension: Particle in a Box

Consider a particle of mass m travelling along x-axis,

confined to a length L by infinite potential barriers at x = 0

and x = L (infinitely deep potential well)

V = V =

V = 0 For x < 0 and x > L,

= 0 since

probability of finding

the particle in these

regions is zero.


9

Particle in one-dimension box

Boundary conditions must be

satisfied by the wavefunctions

9

= sin kx or cos kx or

a combination of both

Bead free to slide

between two stops

QM: wave associated

with particle


10

Particle in 1-D box: Wavefunction

Applying the boundary condition at x = 0 and L gives

Let the trial wavefunction be kxBkxAx cossin)(

0cossin0)(

000cos0)0(

kLBkLAL

BkB

So, we obtain: 0sin kLA

Now A 0; otherwise will be 0 everywhere.

Hence, ,3,2,1,0sin nnkLkL

n cannot be 0


11

The wave function for the system will be

,3,2,1sin)( nxL

nAx

Particle in 1-D box: Normalized wavefunction

Now, if (x) is normalized wave function then

L

dxxx0

)()(*

Solution gives A2L/2 =1 or A = (2/L)1/2

1)sin(*)sin(0

L

dxL

xnA

L

xnA

1sin0

22 L

dxL

xnA

x

L

n

Lx

sin

2)(Normalized wave function for particle

in a 1-D Box


12

Particle in 1-D box: Calculating Energy

Ĥ = E

)(2 2

22

xdx

d

m

x

L

n

Lx

sin

2)(

Quantized energies of the paricle in a 1-D box; n: quantum number

2

22

8mL

hnEn n = 1, 2, …


xL

n

LmL

nh

xL

n

dx

d

LmL

n

sin2

8

cos2

2

2

22

2

x

L

n

Ldx

d

m

sin

2

2 2

22

)]([8 2

22

xmL

nh

13

2

22

18

1

mL

hEn

x

LL

1sin

21

Ground State wavefunction and Energy

If ‘n’ = 0; then = 0 everywhere and particle

vanishes (Not possible).

The energy at lowest cannot be zero;

consequence of uncertainty principle: particle

is within the box, so uncertainty in position, x cannot be infinite; px (or energy) cannot

be zero.


Ground state energy: h2/8mL2 :Zero point energy n

L

L

nhhp

m

p

mL

hnEn

2

2

28

2

2

22

14

Plot of wavefunctions vs position

= L = 2L

= L/2 = 2L/3

= 2L/n; where n = 1,2,3,4……

x

LL

1sin

21

x

LL

2sin

22

x

LL

3sin

23

x

LL

4sin

24

• The wavefunctions are all symmetric or antisymmetric about the

midpoint of the box BITS Pilani, Pilani Campus

15

Probability Amplitude (b) & Density (c)

2

22

18

1

mL

hE

12

22

2 48

2E

mL

hE

12

22

3 98

3E

mL

hE

node

n – 1 nodes in n, energy increases with increasing

number of nodes, no nodes in ground state


16

Classical Limit

16

(iii) As n becomes very large, the probability

distribution becomes uniform (continuous)

over the entire length of the box.

(i) E depend on ‘L’, the size of the system and ‘m’,

the mass of the particle.

(ii) Greater the value of L or m, the less important

are the effects of quantization on translational

motion: classical limit


2

22

8mL

hnEn

At sufficiently high energies, the quantum

mechanics agrees with classical mechanics.

Correspondence principle

BITS Pilani, Pilani Campus 17

Particle in two dimensional (2-D) box

A two-dimensional square

well.

Potential energy is zero

between x = 0 and x = L1 and

y= 0 and y = L2,

Rises sharply to infinity at the

walls.

Ĥx X(x) = Ex X(x)

Ĥy Y(y) = Ey Y(y) E= Ex + Ey

)().(),( yYxXyx

BITS Pilani, Pilani Campus 18 18

Particle in 2-D box: Schrödinger Equation

11

2

L

xnsin

LxX x

22

2

L

xnsin

LyY

y

),()(),(2 2

2

2

22

yxEEyxdy

d

dx

d

myx

On Solving

)().(),(, yYxXyx nynxnynx

2

1

22

8mL

hnE x

x …………………..

………………….. 2

2

22

8mL

hnE

y

y

)()()()()(2 2

2

2

22

yYxXEEyYxXdy

d

dx

d

myx


Particle in 2-D box: Wavefunction & Energy

Where nx= 1,2,3….; ny=1,2,3…

19

2121

sinsin22

),(L

yn

L

xn

LLyx

yx

2

2

22

2

1

22

88 mL

hn

mL

hnE

yx

Now, when L1 = L2

2

222

8

)(

mL

hnnE

yx

nx= 1 & ny=1 : Ground state energy of particle in 2-D box (square)

nx= 1, ny=2 & nx= 2, ny=1 : Degenerate Energy level (Double degenerate states)


Particle in 2-D box (square): Wavefunctions

E1, 1

E2, 1 E1, 2

E2, 2

Doubly Degenerate states

2-fold degenerate

Non Degenerate state

Non Degenerate state

L

y

L

x

Lyx

sinsin

2),(1,1

L

y

L

x

Lyx

2sinsin

2),(2,1

L

y

L

x

Lyx

sin

2sin

2),(1,2

L

y

L

x

Lyx

2sin

2sin

2),(2,2

2

2

1,18

2

mL

hE

2

2

1,22,18

5

mL

hEE

2

2

2,28

8

mL

hE

Notice

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