The Energy Levels of a Rigid Rotator Are E=Ħ2 Ɩ (Ɩ+1)/2I
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical ThermodynamicsLecture 13
The Energy levels of a Rigid Rotator Are E=ħ2 Ɩ (Ɩ+1)/2I
-In this section we shall discuss a simple model for a rotating diatomic molecule.
-The model consists of two point masses m1and m2 at fixed distance r1 and r2 from their center of mass (Fig.1)
-Because the distance between the two masses is fixed, this model is referred to as the rigid rotator model.
Figure (1). Two masses m1 and m2 shown rotating about their center of mass.
-We can treat a rigid rotator as having one mass fixed at the origin with another mass, the reduced mass µ, rotating about the origin at a fixed distance r.
-We discussed a rigid rotator classically before and showed there that the energy of a rigid rotator is
(1)
where ω is the angular velocity and I is the moment of inertia,
I = µ r2 (2)
The angular momentum L is
L = I ω (3)
And the kinetic energy can be written
(4)
The Hamiltonian operator of a rigid rotator is just the kinetic energy operator and using the correspondence between linear and angular systems, we can replace m by I and write it as:
(5)
Because one of the two masses of therigid rotator is fixed as the origin we shall express ?2 is spherical coordinate and so write Ĥ as :
(6)
There is no term in Ĥ here involving the partial derivative with respect to r because r is fixed in the rigid rotator model. By comparing equation (6) with the classical expression equation (4) we see that:
(7)
Note that the square of the angular momentum is a naturally occurring operator in quantum mechanics
-The rigid rotator wave function are customarily denoted by Y(,) and so the Schrödinger equation for a rigid rotator reads
(8)
Or
(9)
If we multiply equation (9) by Sin2() and let
We find the partial differential equation
(10)
β must obey the condition:
β=Ɩ (Ɩ +1) Ɩ =0,1, 2… (11)
Using definition of β, equation 11 is equivalent to
Ɩ =0,1, 2… (12)
The Rigid Rotator Is a Model for a Rotating Diatomic Molecule
ΔƖ = ± 1 (13)
-Equation 13 is called a selection rule
-In the case of absorption of electromagnetic radiation, the molecule goes from a state with a quantum number Ɩ to one with Ɩ +1. The energy difference then is
ΔE = EƖ+1 – EƖ = (14)
-The energy levels and absorption transitions are shown in Figure 2 using Bohr frequency conditions E=h , the frequencies at which the absorption transitions occur are
=0,1,2…. (15)
It is common practice in microwave spectroscopy to write equation 15as
(16)
where Hz is called the rotational constant of the molecule.
If we use a relation
Where is the rotational constant expressed in units of wave numbers?
Cm-1 (17)
Figure 2: The energy levels and absorption transitions of a rigid rotator.
Example
To a good approximation, the microwave spectrum of H35Cl consists of a series of equally spaced lines separated by 6.26×1011Hz. Calculate the bond length of HCl.
Solution:
According to equation (16) the spacing of lines in microwave spectrum is given by
Solving this equation for I
The reduced mass of HCl is
Using the fact that I=r2 we obtain
[1]