Lecture17 Spectroscopy

8/10/2019 Lecture17 Spectroscopy

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Linear Harmonic Oscillator

Energy levels, wave functions,

zero energy level



Objectives

The most important example of elementary Quantum Mechanics

- the linear harmonic oscillator describes vibrations in moleculesand their counterparts in solids, the phonons

-many more physical systems can, at least approximetely, be

described using this model

-the modes of the electromagnetic field (photons- bosons)

provide the conceptual building blocks of microscopic physics

(field quantization)

-string theory (Great Unification Theory in physics)



The Schrödinger equation for the harmonic oscillator

),(),( t x H t x

t

ih Ψ

∂

∂

x

V(x)

The stationary solutions have the form

where is determined by

)()()](2

[ 2

2

x E x x V m Φ

h

The time-dependant Schrödinger equation:

V(x)=1/2Kx2



0)()()(

)()(]2

1

2[

2

2

2

222

=

Φ

ξ

ξd

d

x E x Kx m

h

where

ξ=αx (dimensionalless)

α4=4π2

mK/h2,

λ−dimensionless eigenvalue

λ=(4πE/h)

(m/Κ)1/2=4πΕ/hωc

ωc

= (K/m)1/2

is the angular frequency

of the corresponding classical

harmonic oscillator

From the theory of differential equations the exactsolution can be of the form (Sommerfeld 1929):

2

2

1

)()(ξ

ξ

=

e H

It is satisfying:

-dominant behavior of φ

in the asymptotic

region ξ->+/-~ , n has to be finite-the boundary conditions (the integral of

probability function has to convergeH(ξ) is polynomial of finite order n in ξ (ξn)

The Schrödinger equation is modified to

achieve a convinient mathematical form:



00...)()(0210

≥s a a a a H s

ξ

Solutions are found assuming H in the form:

Assumption:

The equation (*) is to be valid for all values of ξ, so that, once we substitude

H into the equation, the coefficient of each power of ξ

can be equated to zero

0)1(2)( '''

=H H H λ

(*)

s(s-1)a0=0

(s+1)sa1=0

(s+2)(s+1)a2-(2s+0+1-λ)a0=0

(s+2)(s+1)a2-(2s+2+1-λ)a1=0

...........................................

(s+ν+2)(s+ν+1)aν+2

-(2s+2ν+1-λ)aν=0

s=0, s=1

s=0 or a1=0 or both

Then reccurence solution can

be derived

ν

is an integer

(**)

Substitution to the Schrödinger equation:



Discussion of the validity of the calcualted coefficients

1. H(ξ) can be chosen to be either even or odd in ξ

(see particle in the box problem and the parity discussion)

and therefore a1 and all the other odd-subsript coefficients are zero.

The wave function is then even or odd, accordingly (s=0 or s=1; see (**))

2. The existance of a finite or infinite number of terms depends on the choice ofs and the λ (

is related to eigenvalue E/hωc

). The series (**) must terminate

and therefore λ=2s+2ν+1.

For s=0 or 1, one obtains λ=2ν+1 or 2ν

+ 3 and this value is related

to even or odd wave function.

Both cases can be expressed in terms of quantum number n:

...,2,1,0)2

1(12 =n n E n

c

n ω

λ

h



Wave functions

22

2

1

)()(x

n n n e x H N

=

The quantum number n is the highest value of s+ν

in the series for H. It

will be denoted that Hn is of degree n in ξ

and is even or odd according

as n is even or odd.

Hn(ξ

) is known as nth Hermite polynomial

The corresponding eigenfunction

φn

(ξ) has the parity of n and

has n nodes.

2)(4)(2)(1)( 2

210 x x H x H x H

First three memebers are of the form



There is a convinient formulation which expresses the Hn in termsof a generating function:

n n

n

s s s

n

H e s S

!

)(),(

0

22

ξ

ξ

ξ

∑

=

=

This enables to obtain an expression for the nth Hermite polynomial:

22

)1()( ξ

ξ

ξ

∂

∂

e e H n

n n

n

Generating function for the Hermite polynomians

Other interesting relationships are:

11

1

22

2'

=

n n n

n n

nH H H

nH H

ξ



Orthogonality and normalization

of the harmonic oscyllator wave functions

1)()(2

2

2

2=

∫

∞

∞

∞

∞

ξ

ξ

d e H N

dx x n

n

n

ξ

ξ

d e H H m n

t s

d e e e m n

m n

m n

t t s s 2222

)()(!!00

22

∞

∞

∞

=

∞

=

∞

∞

∫

=

One can easily prove that the harmonic oscillator wave functions are orthogonal

and the normalization constant Nn can be obtained from normalization procedure:

The generating function is useful for the calculations of integrals involving

the harmonic oscillator wave functions.

The integral on the right can be expressed as a series coefficient in the expansion

of an integral containing the product of two generating functions

∑

=

=

0

2/122/1

!

)2(

n

n st

n

st e π

And one has to equate powers of s and t in series

on the right side.



2/1

2/1 )

)!2((

n N

n n π

=

!2)( 2/12

2

n d e H n

n π

ξ

=

∞

∞

m n d e H H m n ≠

∞

∞

∫

0)()(

2

ξ

ξ

From the first equation one can deried that the normalizing constant can be chosen:

orthogonality



Stationary states φ(x)

of the harmonic oscillator for n=0, 1, 2, 3, 4

x

22

2

1

22/1

2/12 ]2)(4[)2

()(x

e x x

π

22

2

1

2/1

2/10 )()(x

e x

π

=

22

2

1

2/12/11 2)

2()(

x

xe x

π

=



Properties of the solution:

1. There is infinite sequence of energy levels and the equal spacing between them

2. The ground state energy has the finite value 1/2h/ωc —zero-point energy— and is related to the uncertainty principle

22

)()(

x K m

p Δ

the total energy

Δp, Δx –are measures of the spreads in the momentum

and position, and

Minimizing this energy by taking the derivative with respect to the position x

energy and setting it equal to zero gives and minimum value of the total

energy allowed:



This is a very significant physical result because it tells us that the energy of a

system described by a harmonic oscillator potential cannot have zero energy.

Physical systems such as atoms in a solid lattice or in polyatomic molecules in a

gas cannot have zero energy even at absolute zero temperature. The energy of

the ground vibrational state is often referred to as "zero point vibration".

The zero point energy is for instance sufficient to prevent liquid helium-4 from

freezing at atmospheric pressure, no matter how low the temperature.

The ground state energy has the finite value ½ hωc



3. Correspondance with classical theory

The position probability densities associated with these stationary wave functions

have a little resemblance to the corresponding densities for classical harmonic

oscillator for low n but for large n the agreement is fairly good on the average

(discrepancy: rapid oscillations).

The expectation values:

-the expectation value for the potential energy:

(En-is the total energy)

-the expectation value for the position and momentum

<x>=<p>=0 for any harmonic –oscillator wave function

4. The uncertainty product Δx Δp=(n+1/2)h/2π and it is ½

h/2π for the

ground state

eigenfunction

n c n n n E n

n K dx x Kx V

2

1)

2

1(

2

1

2

12

2

1)(

2

12

2*==

∫

∞

∞

ω

h

It is of the form of the minimum packet.



Picture of wavefunctions (and energy levels)

of quantum harmonic oscillator

Lecture17 Spectroscopy

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