Question

Derive expressiom for various energy levels of a rigid rotator

Answer 1

Answer:

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Explanation:

The rigid rotor is a mechanical model that is used to explain rotating systems. The linear rigid rotor model consists of two point masses located at fixed distances from their center of mass. The fixed distance between the two masses and the values of the masses are the only characteristics of the rigid model. However, for many actual diatomics this model is too restrictive since distances are usually not completely fixed and corrections on the rigid model can be made to compensate for small variations in the distance. Even in such a case the rigid rotor model is a useful model system to master.

For a rigid rotor, the total energy is the sum of kinetic ( T ) and potential ( V ) energies

Etot=T+V(5.8.1)

The potential energy,  V , is set to  0  because the distance between particles does not change within the rigid rotor approximation. However, In reality,  V≠0  because even though the average distance between particles does not change, the particles still vibrate. The rigid rotor approximation greatly simplifys our discussion.

Since  V=0  then  Etot=T  and we can also say that:

T=12∑miv2i(5.8.2)

However, we have to determine  vi  in terms of rotation since we are dealing with rotation motion. Since,

ω=vr(5.8.3)

where  ω  is the angular velocity, we can say that:

vi=ωXri(5.8.4)

Thus we can rewrite Equation  5.8.2  as:

T=12∑mivi(ωXri)(5.8.5)

Since  ω  is a scalar constant, we can rewrite Equation  5.8.5  as:

T=ω2∑mi(viXri)=ω2∑li=ωL2(5.8.6)

where  li  is the angular momentum of the ith particle, and  L  is the angular momentum of the entire system. Also, we know from physics that,

L=Iω(5.8.7)

where  I  is the moment of inertia of the rigid body relative to the axis of rotation. We can rewrite Equation  5.8.2  as

T=ωIω2=12Iω2(5.8.8)

Equation  5.8.8  shows that the energy of the rigid rotor scales with increasing angular frequency (i.e., the faster is rotates) and with increasing moment of inertia (i.e, the inertial resistance to rotation). Also, as expected, the classical rotational energy is not quantized (i.e., all possible rotational frequencies are possible).

The Quantum Rigid Rotor in 3D

Answer 2

The Energy Levels of a Rigid Rotor :

A rigid rotor is one in which the spacing between the individual particles remains constant while it rotates. If vibration is disregarded, a rigid rotor can only roughly simulate a revolving diatomic molecule.

3D version of the traditional rigid rotor:

• A mechanical paradigm for explaining rotating systems is the stiff rotor.
• Two-point masses are used in the linear rigid rotor model, and they have spaced a certain amount apart from one another.
• The only traits of the rigid model are the fixed separation between the two masses and the mass values.
• However, since distances are typically not entirely set and changes can be made to the rigid model to account for slight variations in the distance, this model is too limiting for many real diatomic.
• The rigid rotor model is a helpful model system to master even in this situation.
• The sum of the kinetic (T) and potential (V) energies for a stiff rotor represents the total energy.

$E_{tot} = T + V$  (5.8.1)

• The rigid rotor approximation assumes that the distance between particles does not change, hence the potential energy, V, is set at 0.
• Although the average distance between particles does not change, the particles still vibrate,

hence in actuality, V0.

• Our discussion is substantially streamlined by the rigid rotor approximation.

Since V=0, $E_{tot}$=T follows, and we can further state that:

$T = \frac{1}{2}$∑$m_{i}v^{2}_{i}$ (5.8.2)

• However, because we are dealing with rotation motion, we must determine vi in terms of rotation.

Since,

$ω = \frac{v}{r}$ (5.8.3)

where the angular velocity is, the following is true:

$v_{i}= ωXr_{i}$ (5.8.4)

• Thus, Equation 5.8.2 can be rewritten as follows:

$T=\frac{1}{2}$∑ $m_{i}v_{i}(ωXr_{i} )$  (5.8.5)

• Since is a scalar constant, Equation 5.8.5 can be rewritten as follows:

T = $\frac{ω}{2}$∑$m_{i}(v_{i}Xr_{i} )$ $l_{i} = ω\frac{L}{2}$ (see 5.8.6),

where li is the angular momentum of the with particle and L is the system's overall angular momentum.

• In addition, physics has taught us that

L=Iω (5.8.7),

where I is the stiff body's moment of inertia with respect to the axis of rotation.

• We can change Equation 5.8.2 to read as

T=ω$\frac{Iω}{2}$=$\frac{1}{2} Iω^{2}$ (5.8.8)

Formula 5.8.8

• Demonstrates that the rigid rotor's energy increases with both moments of inertia and angular frequency, or how quickly it revolves (i.e, the inertial resistance to rotation).

The classical rotational energy is also not quantized, as expected.

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