The Schrodinger Equation Chapter 2 Quantum Chemistry and Spectroscopy

Ch 2. The Schrödinger Ch 2. The Schrödinger Equation (S.E)Equation (S.E)

MS310 Quantum Physical Chemistry

- Due to the contribution of wave-particle duality, an - Due to the contribution of wave-particle duality, an appropriate wave equation need to be solved for the appropriate wave equation need to be solved for the microscopic world.microscopic world.

- Erwin Schrödinger was the first to formulate such an Erwin Schrödinger was the first to formulate such an equation equation

- We need to be familiar with operators, eigenfunction, - We need to be familiar with operators, eigenfunction, wavefunction, eigenvalues that are used in S.E.wavefunction, eigenvalues that are used in S.E.


2.1 What determines if a system needs to be described using Q.M?

When do we use a particle description(classical) of an atomic or molecular system and when do we use a wave (quantum mechanical) description? two criteria are used!

1) The magnitude of the wavelength of the particle relative to the dimension of the problem, a hydrogen molecule 100 pm vs. a baseball 1x 10-34 m no way to show wave character of baseball

2) The degree to which the allowed energy values form a continu ous energy spectrum

kT

j

i

j

i jieg

g

n

n /][ Boltzmann distribution :


There are two limits

1) large T or small εi – εj

: small energy gap(almost continuous energy),

classical behavior

2) (εi – εj)/kT >> 1

: large energy gap, quantum behavior

Also we can derive the internal energy by Boltzmann distribution

k : the Boltzmann constantkTE2

3


Relative population in the different energy levelsa) Sharp energy levelsb) ∆E ≈ kT : classical behavior(nearby the continuous energy)c) ∆E >> kT : quantum behavior(discrete energy)

2.2 Classical waves and the nondispersive wave equation

Example of waves

a) Plane wavesb) spherical wavesc) cylindrical waves

Wave front : surface over the maximum or minimum amplitudeDirection of propagation of waves : blue arrows, perpendicularto the surface



T

tnxn

T

tx

22

T

txAtx

2sin,

Arbitrary condition : Ψ(0,0)=0

Positions where amplitude is zero :

x or t increase → wave moves in the positive x direction

Mathematically, we can describe the wave by wave function.Amplitude of wave : related to position and timePosition : propagation depends on wavelength λTime : propagation depends on period T

Graph of wave functions


)sin(, tkxAtx

We use another form, too.

k : wave vector,

2k

ω : angular frequency, ω=2πν

Choice of a zero in time or position : arbitrary(free)

→ We can rewrite the wave equation

)sin(, tkxAtx

In this case, Ψ(0,0) ≠ 0 and φ : initial phase


If two or more waves in same region : interference

1) constructive interference : enhancement of amplitude 2) destructive interference : cancellation of amplitude

Phase difference in interference 1) constructive interference : 2nπ → same sign 2) destructive interference : (2n+1)π → opposite sign

Also, we can think about two waves same frequency,amplitude and opposite directions

)sin(),( tkxAtx

)sin(),(2 tkxAtx

)]sin()[sin(),(),(),( 21 tkxtkxAtxtxtx


Use this formula

sincoscossin)sin(

Final result is simple than original equation

txtkxAtx cos)(cossin2),( position of nodes are same at all time!→ ‘standing’ wavesstanding waves represent the stationary state(no change by time)


Both of traveling waves and stationary waves, amplitudeand distance is not independent

2

2

22

2

t

)t,x(Ψ

v

1=

x

)t,x(Ψ

∂∂

∂∂

: classical nondispersive wave eq.

v : velocity of wave propagationThis equation is a start point of the Schrödinger Equation.

Ex) 2-2. show that traveling wave satisfies wave equation.

Solution)

)φ+tωkxsin(Ak=x

)t,x(Ψ 22

2

--∂

∂

)φ+tωkxsin(A=)t,x(Ψ -

)φ+tωkxsin(Av

ω=

t

)t,x(Ψ

v

12

2

2

2

2 --∂

∂

Equating two results, v=ω/k


2.3 Waves represented by complex functions

)'cos()cos()sin(,

tkxAtkxAtkxAtx

Where φ’ = φ - π/2

Use the Euler’s formula sincos)exp( iie i

))'(expRe(, tkxiAtx

)'(exp, tkxiAtx

We write the wave equation in complex form

We know the traveling wave equation

Why uses a complex form? 1) All quantities can obtain by this form 2) easier than real form with differentiation and integration.


Important formulas

1) Complex number : a+ib(a,b:real) or equivalent form,

2) Complex conjugate of f(number of function) : f* substituting i to –i ex) (a+ib)* = a-ib,

3) Magnitude of f(number of function) : |f|

)cos and( -122

r

abarre i

ii rere *)(

fff *||

rrebaiba i ||,||ex) 22

2.4 The Schrödinger Equation

How we can obtain the Schrödinger Equation?→ start with the classical wave equation and stationary wave

txtx cos)(),( Substitute the wave equation by stationary wave, we obtain

0=)x(ψv

ω+

dx

)x(ψd2

2

2

2

Using the relations ω=2πν and νλ=v

0=)x(4

+dx

)x(d2

2

2

2

ψλ

πψ

Until now, it is classical wave.



‘Introduce’ the quantum mechanics by de Broglie relationp

h

Momentum is related by total energy E and potential energy V(x)

)((2),(2

2

xVEmpxVEm

p

Substituting the wave equation by this relation

0=)x()]x(VE[h

m8+

dx

)x(d2

2

2

2

ψπψ

-

Use =h/2ℏ π, we obtain the time-independent Schrödinger Eq.

)()()()(

2 2

22

xExxVdx

xd

m

0=)x(h

p4+

dx

)x(d2

22

2

2

ψπψ

How one obtain the time-dependent Schrödinger Equation? Starting from the solution of classical wave equation

xvti

xti

e

Aetx

2

A

),(

On the other hand

22 E hE

212 p

pp

h

2h cf.

pxEti

Aetx

),(Then

Wave equivalent of a free particle ofenergy E and momentum p moving on the x-direction


Let’s try now,

2

2

p

pi

xpi

xx 2

222

xp

Eit t

iti

E

We know that ),(2

2

txVm

pE

),(),(),(2

),(2

),(

2

2

txtxVtxm

p

txVm

ptxE

Now replace ),(p ),,( 2 txtxE

),(),(),(

2

),(2

22

txtxVx

tx

mt

txi

Time-dependentSchrödinger Eq.



One of our focus is stationary system.In this case, both of time-dependent and time-independent are satisfied.

),(),(

txEt

txi

For stationary state, )()(),( tfxtx

Substitute the equation, we obtain

tE

ietftf

Ei

dt

tdftEf

dt

tdfi

)(),(

)(),(

)(

Finally,t

Ei

extx

)(),( → same form as classical standing wave

2.5 Solving the Schrödinger Equation

Key concept : operators, observables, eigenfunctions and eigenvalues

2

1

),(1

)()(

),(

12

2

2

t

t

dttxFm

tvtv

txFdt

xdm

Operators in a classical mechanicsEx) Velocity in Newton’s second law

1) Integrate the force acting on the particle over the interval2) Multiply by the inverse of the mass3) Add the quantity to the velocity at time t1



How about a operators in Quantum Mechanics?

→ every measurable quantities(observables) have their operator each.(ex : energy, momentum, position)Notation : caret, Ô

Differential equations : set of solutionsOperator Ô has a set of eigenfunctions and eigenvalues

nnn a O

Ψn : eigenfunctions, an : eigenvalues

Ex) hydrogen atomeigenfunctions : each orbitals(1s, 2s, 2px, …)eigenvalues : each orbiral energies


We see the time-independent Schrödinger equation.

)()()}(2

{ 2

22

xExxVxm nnn

HxVxm

ˆ )amiltonianoperator(H energy : )(2 2

22

nnn EH ˆThis equation can be written by

Important physical implication : measurement process in Q.M.

2.6 Eigenfunctions of Q.M. operator are orthogonal

Orthogonality in 3-dimensional vector space : x•y = x•z = y•z = 0

Similar, orthogonality in functional space is defined by

ji unless,0)()(* dxxx ji

operator quantum of ionseigenfunct :)(),( xx ji

If I = j, the integral has a nonzero valueFunctions can be normalized and form an orthonormal set.

nidxxx ii ,...,2,1,1)()(*



In 3-dimension, normalization must be 3-dimension.

Closed-shell atoms are spherical symmetric and we normalized the wave functions in spherical coordinate

Volume element in spherical coordinate : r2 sin θ dr dθ dφ, not a dr dθ dφ

2.7 Eigenfunctions of Q.M. operator form a complete set

completeness in 3-dimensional vector space :Any vector in 3-dimensional can be represented by linear combination of vector x, y, and z

Similar, completeness in functional space : Wave function can be expanded in the eigenfunctions of any Q.M. operator

1

)()(n

nn xbxf

It is same formalism as a Fourier series.

We choose a periodic function in [-b,b]Fourier series is represented by

1

0 )]cos()sin([)(n

nn b

xnd

b

xncdxf



If f(x) even : cn=0, dn calculated by orthogonality

dxb

xmdd

b

xmdx

b

xmxf

mn

b

b

b

b

))(cos())(cos()cos()( 0

b

b

mm bddxb

xmd

b

xm)cos())(cos(

Generally

dxxfb

d

mdxb

xmxf

bd

b

b

b

b

m

)(2

1

0,)cos()(1

0

And this approximation is nearly exact.

m

nn b

xnddxf

10 )cos()(


Accuracy of Fourier series a) Yellow line : real function

b) red line : fourier series approximation, n=2,3,4 and 6

11,}1)(2{)(2

2

22

b

xe

b

xxf b

x


- The time-dependent and time-independent Schrödinger equations play the role in solving quantum mechanical problems.

- Operators, eigenfunctions, wave functions and eigenvalues are key concepts to solve quantum mechanical wave equations.

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The Schrodinger Equation Chapter 2 Quantum Chemistry and Spectroscopy