BioPHYSICAL chemistry

so equivalently:

ψ(x) =Aeikx (9.42)

Substitution into eqn 9.39 gives:

(9.43)

(9.44)

(9.45)

(9.46)

Born interpretation

The physical interpretation of quantum mechanics, in particular the

interpretation of the wavefunction, was developed by many scientists,

most notably Max Born (Nobel Prize winner in Physics in 1954). Since

all particles are also waves, particles are always distributed in space. The

wavefunction in Schrödinger’s equation has no direct physical meaning

and can be a complex function rather than a real function. In the Born

interpretation, the probability of finding any particle at a particular loca-

tion is not given by the wavefunction itself but rather by the square of

the wavefunction. These ideas led to several fundamental postulates of

quantum mechanics, as follows.

1 A particle is never at a specific location but only has a probability of

being there. The probability of finding a particle at a specific position

is given by the square of the wavefunction times the volume dτas:

ψ*(r)ψ(r)dτ

where dτ=dx dy dz=r^2 sinθdr dθdφ (9.47)

In this equation ψ*is the complex conjugate. Since any wavefunction

ψcan be written in terms of two real functions, Aand B, the conjugate

can be defined as:

ψ=A+iB and ψ* =A−iB (9.48)

The probability of finding a particle within a volume Vis then:

(9.49)

where ψ*(x) is the complex conjugate of the wavefunction.

ψψ τ*d() ()rr

V

∫

→= =pk =

hh

Z

2

2

π

π

λλ

EV

k

m

p

m

−= = =

Z^222

22

kinetic energy

()Ae E ()()()()

m

ikx =− −kAeikx +VrAeikx

Z^22

2

EAe

mx

()ikx =− ()()()Aeikx +V r Aeikx

Z^22

2 2

∂

∂

188 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY