Download Quantized Vibrational Energy for a diatomic molecule

Where do the energy equations come from?
The motion of atoms, molecules, electrons … is described by Quantum Mechanics.
The central equation of Quantum Mechanics is the Schrödinger Equation. Solving the
Schrödinger equation for a ‘problem’, results in an expression for the energy of the
particle(s) in the problem and a wavefunction.
The wavefunction is a mathematical function of the coordinates of the particle(s) and
provides information on the location of the particle(s).
(Note: In Classical Mechanics, one solves Newton’s 2nd Law equation to get the energy
and position of the particle(s).)
A prescription for obtaining the Schrödinger equation of a problem for one particle of
mass m, moving in 1D x.
1)
H ( x) ( x)  E ( x)
is the completely general form of the time-independent Schrödinger equation
where
is the wavefunction, a function of the coordinate x
 (x)
E
H (x)
is the Energy of the particle
is the Hamiltonian operator
(Note: want to determine  (x) and E by solving the equation.)
2) Write the Hamiltonian in Classical terms as the sum of Kinetic Energy (T) and
Potential Energy (V)
H ( x )  T ( x )  V ( x )  p / 2m  V ( x )
2
(Note: T = mv2 / 2 = p2 / 2m and V(x) depends on the problem being considered.)
3) Make the change from Classical Mechanics to Quantum Mechanics by introducing:
p  i 
p   
2
2
2
2
d
 
dx
2
2
Note: V(x) seldom (never in this course) depends on p and therefore this change only
affects the p2 / 2m term.
H ( x) ( x)  E ( x)
   d 

 2m  dx   V ( x)  ( x)  E ( x)
2
2
2
This is the time-independent one dimensional Schrödinger Equation
Note: Hamiltonian operator because it involves a differential operator (d2 / dx2). The
‘relationship’, p  i , is the essential difference between Classical and Quantum
Mechanics. The operator means that there has to be something, a function, for it to
operate on – the wavefunction.