Quantum Mechanical Treatment of a Rigid Rotor

Rigid Rotator (or rotor):

Ignoring masses of electrons & the vibration of nuclei, a diatomic molecule may be taken as a rigid rotor. It is assumed that rotation involves no change in the centre of gravity and the bond length (r). If the centre of gravity be at the origin of the coordinate system and r₁ & r₂ be the distance of m₁ & m₂ from the centre of gravity and two masses m₁ & m₂ move with linear velocity ν₁ & ν₂ respectively then rotation K.E. is given by—

K.E. =

1

2

m₁ ν₁² +

1

2

m₂ ν₂² =

1

2

m₁ ω² r₁² +

1

2

m₂ ω² r₂²

=

1

2

ω² (m₁r₁² + m₂r₂²) =

1

2

ω²I =

L²

2I

where ω = angular velocity and I = moment of inertia about an axis passing through the centre of gravity and normal to the line through the masses and L (i.e. ωI) = total angular momentum of rotation.

Rotation of a rigid diatomic molecule can occur in a plane i.e. with fixed rotation axis or in space i.e. with free rotational axis.

Rotation in plane:

If a diatomic molecule rotates in a XY plane, the rotational axis coincides with Z-axis. The angular momentum will have only the Z-component L_z (L_x = L_y = 0) because the molecule cannot move out of the plane). The rotational energy of the molecule will be equal to K.E. i.e. ν = 0, hence

Ĥ =

L_z²

2I

= –

h²

8π² I

d²

dφ²

... (1)

and the Schrödinger equation—

d²ψ

dφ²

= –

8π²μ E

h²

ψ = – M²ψ

where φ is the angle of rotation in a plane and M is the reduced mass i.e.

m₁m₂

m₁ + m₂

, eigen functions corresponding to eq. (1) are known to be

ψ =

1

√2 π

e ^{i M φ}

and the eigen values are—

E =

M²h²

8π² I

     or, M = 0, ± 1, ± 2 ...

where      I = Mr²

In other words, a diatomic particle rotating in a plane may be taken as a single particle.

Rotation in space:

Rotation in space means that the axis of rotation is free to take any direction in space. In this case, there are two angular variables θ & φ; θ is the angle which the axis of rotation makes with the Z-axis and φ is the angle in the xy plane between the x-axis and the projection of the rotational axis in the xy plane, since the inter-nuclear distance does not change.

The P.E. (V) is zero—

∴      Ĥ =

L̂²

2I

∴      L̂² = –

h²

4π²

[

1

sin θ

·

∂

∂θ

( sin θ ·

∂

∂θ

) +

1

sin² θ

·

∂²

∂φ²

]

Hence Schrödinger equation Ĥ ψ = E ψ may be written as—

1

sin θ

·

∂

∂θ

sin θ ·

∂ψ

∂θ

+

1

sin² θ

·

∂²ψ

∂φ²

+

8π²IEψ

h²

= 0

Let ψ = Θ (θ) Φ (φ), then—

sin θ

Θ

∂

∂θ

( sin θ ·

∂ Θ

∂θ

) +

8π² I

h²

sin² θ = –

1

Φ

∂²Φ

∂φ²

Setting both sides to a constant, m² (say) we have two differential equations, each in one variable as—

∂²Φ

∂φ²

+ m²Φ = 0 ... (1)

1

sin θ

·

∂

∂θ

( sin θ ·

∂ Θ

∂θ

) + [ β –

m²

sin² θ

] Θ = 0 ... (2)

where      β = 8π² IEh²

Solution of eq. (1) is Φ (φ) = C . e ^{± i m φ}

This is an acceptable wave function provided m is an integer. This condition arises because Φ is required to be single valued—

Φ (φ) = Φ (2π + φ)

or,

e ^{i m φ} = e ^{i m (φ + 2π)}

This requires e ^{2π mi} to be unity i.e.

cos 2π m + i cos 2πm = 1

This is true if m = 0, ± 1, ± 2, ± 3 .... etc.

The normalisation condition gives C the value 2π^{– 1/2}. The normalised solutions are—

Φ _{± m} (φ) =

1

√2π

e ^{± im φ}      (m = 0, 1, 2, 3....)

Eq. (2) has its solutions the associated Legendre polynomials P_l^|m|(cos θ) where l is either zero or a positive integer and further l ≥ | m |. The normalised solutions are given by—

Θ (θ) = Θ _{l, ± m} (θ) = √

2l + 1

2

·

(l – | m |)!

(l – | m |)!

P_l^|m|(cos θ)

The restriction on l leads to quantisation of K.E. of rotation whose values are given by—

E =

l(l + 1) h²

8π² I

The eigen functions ψ (θ, φ) are the spherical harmonics.

A diatomic molecule rotating in space with free axis may therefore be treated as a particle on the surface of a sphere.

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Conclusion & Physical Significance of Quantum Numbers:

• l (Rotational Quantum Number): Where l = 0, 1, 2, 3... This determines the total magnitude of the rotational angular momentum.
• m (Magnetic Quantum Number): Where m = 0, ±1, ±2... ±l. This determines the orientation or projection of the angular momentum vector along a fixed reference axis (usually the Z-axis).