LECTURES ON QUANTUM MECHANICS

by Reinaldo Baretti Machín (reibaretti2004@yahoo.com)

Chapter 6 - Periodic Potentials

Lectures on Quantum Mechanics -Introduction

Chapter 1 - Some experimental facts

Chapter 2 - The Old Quantum theory

Chapter 3 - The Postulates of Quantum Mechanics

Chapter 6 - Periodic Potentials

Chapter 7 - Angular Momentum

Chapter 8 - The Hydrogen Atom

FREE DOWNLOAD FORTRAN-FORCE 2.0.8

References:

1. Principles of quantum mechanics; nonrelativistic wave mechanics with illustrative applications.

2. Introduction to Solid State Physics (Hardcover) by Charles Kittel

Electron in metals can be treated to a first approximation as entirely free , except for the box confinement which is represented by an infinite wall potential. In this model the interaction between the free electrons is neglected as well as the interaction between the electron and the sea of positive ions.

The allowed energies are those
already known from the infinite box E_{n} = (1/2) (n π /L_{box}
)^{2} where L_{box} = N d ; N being the number of atoms and d
the separation in a linear chain.

The next best approximation would be to suppose that in a linear chain of N atoms separated a distance -d- there are N electrons , if each atom contributes one electron to the free electron sea. Or instead one may suppose that there are 2N electrons if each atom allows two electrons to roam freely. The electrons interact with the periodic potential of the ions.

We integrate numerically Schrodinger equation using the FORTRAN code given below. A linear chain of "6 atoms " is simulated with the potential

v=v0*(1-sin(pi*x/d)^2) , v0=10 , d= 1 .

The initial conditions on Ψ are Ψ (0)=0. ,
(dΨ/dx)_{x=0} =1 .

Sage math

# plot periodic potential

v0=10 ; d=1 ; v=v0*(1-sin(pi*x/d)^2) ;

y=plot(v,x,0,6*d)

show(y)

Fig 1 . Periodic potential V=v0*(1-sin(pi*x/d)^2) ,
v0=10, d=1, with boundaries at x=0, x=L_{box}
= 6d .

At the boundaries V(x)= ∞ , for x < 0 , x > L .

V(x+d) =V(x) d is the atomic separation.

**Bloch waves**

A wave function is of the form

Ψ(x) = u(x) exp(ikx) (1)

It is a wave function for a chain
of N atoms separated a distance d. The size of the box is Nd=L_{box} .

It is assumed that Ψ(x) = Ψ(x
+ n L_{box}) . Then , in a mathematical sense the box is infinite with
period L_{box}. On the other hand

the function u(x) is periodic with respect to lattice translations , u(x + nd) = u(x).

What we will asume here instead is that
the ends of the box represent an infinite potential imposingΨ(0) = Ψ(L_{box})
=0 . Equation (1) will be modified to

Ψ(x) = u(x) sin(kx) . (2)

The trigonometric part sin(kx) takes
into account
the boundary conditions of the box. k =( n π /L_{box} )
n=1,2,3....

The function u(x) is estimated by dividing Ψ(x) /sin(kx) . Of course near the zeros this would give wrong values. Nevertheless the shape of u(x) can be discerned from the plot confirming the periodicity of u(x).

Fig 2. First state E1= 4.50

Fig 3 . Second electron state , E =4.865 ,
k_{2} = (2π /L_{box} )

Fig 4. Second state. Shows
Ψ_{2} and u (x) .The errors
or distortion in u(x) at x=0 , x=3 and x= L_{box} is due to the fact
that we are dviding Ψ_{2} by sin(k_{2}x) to extract u(x) . At
those point the sine function is zero giving invalid results and L'Hopital rule
should be used.

Fig 5. E3 = 5.435 , k_{3} =(3π
/L_{box} ) again there is distortion in u(x) at the
zeros of sin(k_{3} x) .

Fig 6 . E(k) shows a gap at k= π/d .Another gap would show at k= 2π/d , 3π/d...etc .They are less pronounced when E >> V(x). As k becomes larger

E(k) approaches (1/2)k^{2} .

Fig 7. Wave function for the 8-th state
, it is the second state in the second "band". Ψ_{8} is un normalized
and is

practically identical to sin (k_{8}x).
Compare to the ground state wave function in Fig 2.

c Periodic potential /Bloch waves

real Lbox , Lscale ,k1 ,k2 ,k3,k4,k5,k6, k

dimension psinf(100),energy(100)

data niter/1/

data v0,d, natoms /10.,1.,6/

v(x)=v0*(1.-sin(pi*x/d)**2 )

pi= 2.*asin(1.)

Lbox=float(natoms)*d

k1=pi/lbox

k2=2.*pi/lbox

k3=3.*k1

k4=4.*k1

k=k3

e=5.435

efinal=5.70

de=(efinal-e)/float(niter)

xi=0.

xlim=Lbox

lscale=1./abs(v0)**.5

dx=lscale/300.

nstep=int(xlim/dx)

e1box=(1./2.)*k1**2

print*,'e1box,e2box=', e1box,4.*e1box

print*,'xlim,dx,nstep=', xlim,dx,nstep

print*,' '

do 110 iter=1,niter

c dx=xlim/float(nstep)

kp=int(float(nstep)/90.)

kount=kp

c IC for even functions

c psi0=1.

c psi1=psi0

c IC for odd functions

psi0=0.

psi1=psi0+dx

if(niter.eq.1) print 200 ,0.,psi0,psi0/sin(k*(0.+dx)),sin(k*xi)

do 100 i=2,nstep

x=xi+dx*float(i)

psi2=2.*psi1-psi0 -dx**2*2.*(E-v(x-dx))*psi1

if(i.eq.kount)then

uk=psi2/sin(k*x)

if(niter.eq.1)print 200 ,x, psi2 ,uk,sin(k*x)

kount=kount+kp

endif

psi0=psi1

psi1=psi2

100 continue

energy(iter)=e

psinf(iter)=psi2

20 continue

e=e + de

110 continue

do 30 i=1,niter

If(niter.gt.1)print 120 ,energy(i),psinf(i)/abs(psinf(i))

30 continue

120 format(3x,'energy,psinf=',2(3x,e10.3))

200 format(3x,'x,psi,un(x),sin(k*x) =',4(3x,e10.3))

stop

end