rotational motion

what is rotational motion?

motion of a single particle constrained to move around a central point and at a fixed distance from it

what type of motion is rotational? potential?

periodic motion on a flat potential

show 2D and 3D rotational motion

2D- circular motion of a particle on a ring

3D- motion of a particle on the surface of a sphere

can the particle move freely?

yes as there is no potential energy gradient pushing it one way or another

particle doesn’t interact with anything as it moves

• nothing to speed it up or slow it down

potential is constant (flat) = 0

what are the similarities and differences between translation and rotation?

both have motion on a flat potential

rotation has periodic boundary conditions

• after gone all the way around, it is back where its started

• motion is never reversed (no walls to bounce off)

for particle on a ring, show diagram

• what are the fixed values?

• what is the variable value? how does this change after a full cycle?

V potential energy = 0 everywhere

movement east to west

after a full cycle, ϕ has changed by 2π

for particle on a sphere, show diagram

• what are the fixed values?

• what are the variables? how does this change?

at θ=0 you’re at the North Pole, at θ=π you’re at South Pole

what is the solution when you solve the Schrödinger equation for a H atom?

the angular part of the solution is what you get from the particle on a sphere

show rotational motion of a diatomic molecule. show θ,𝜙 and μ_x

what is this called?

two atoms form a rigid rotor

• same as a particle with mass μ rotating at a fixed distance r from the centre of mass

what happens when the rigid rotor absorbs or emits microwave radiation?

undergoes rotational transitions

jumping from one rotational EL to another

show rigid rotor diagram in 2D and show r, x and m

show relevant equations 2

what is angular momentum in simple terms?

how hard it is to stop movement

explain quantisation in rotational motion?

energy and angular momentum are quantised

caused by cyclic boundary conditions

is the ZPE in rotational motion?

no - a rotating particle can stay still

what is angular momentum in classical mechanics?

the vector product of the position vector and the momentum vector

it is perpendicular to the plane of rotation

what is the right hand rule?

thumb points in the direction of angular momentum

what is kinetic energy for a particle constrained to move at a fixed distance r from the origin? how is this similar to translational?

KE directly related to the magnitude of angular momentum

for translational: K=|p|²/2m

• moment of inertia I=mr² plays the role that mass plays in translation

show the angular momentum equations and overall modulus

how is particle in a box, particle on a ring/sphere similar in terms of energy and separation of variables?

for all 3, total energy is kinetic and so separation of variables remains applicable

• potential energy = 0

explain separation of variables

particle in a box - hamiltonian sum of two kinetic energy operators (motion along x and y). because it is a sum of individual terms, separation of variables applies

on a ring - only one variable 𝜙 so there is nothing to separate. hamiltonian is KE operator for motion along 𝜙 direction

on a sphere - two variables. hamiltonian is the sum of 2 kinetic operators (one for each direction)

how does KE separation affect the wavefunction?

because KE separates into independent terms for each variable, the wavefunction can be written as a product of functions (each depending on one variable)

what is the kinetic and p operators for motion along x for particle in a box?

how does this differ from particle on a ring?

KE involves angular momentum

which direction is angular momentum direction?

what is plane of rotation? what does this mean about the angular momentum components?

angular momentum assumed to be z direction

plane of rotation is xy plane

components of the angular momentum along x and y are both zero

what can you say about the uncertainty principle in this case?

components of angular momentum along x and y are zero

however this is forbidden by the uncertainty principle = major reason why this is not used for practical applications

how is this written for particle on a ring? what can be said about it?

angular momentum vector ℓ points along ±z so only need to consider ℓ_z

• magnitude coincides with the magnitude of ℓ

how does this change for particle on a sphere?

ℓ can point anywhere so have to consider all components

angular momentum again replaces linear momentum

what are the boundary conditions for particle on a ring?

no walls so particle moves freely around the circle

there is a constraint - after going all the way around, it must be back where it started

restarts the cycle - angle 𝜙 and angle 𝜙+2π describe the same point

for particle on a ring, what type of boundary conditions is it? what is the period?

periodic/cyclic

wavefunction repeats itself with a period of 2π

what is the periodicity requirement for particle on a sphere?

if you start at a point with particle values of θ and 𝜙, you go to the opposite side of the sphere by flipping θ to -θ, and you get back to the starting point by adding π to 𝜙

what is the consequence of this periodicity?

gives rise to quantisation of angular momentum and therefore quantisation of energy

only certain values of the angular momentum are allowed - those for which the wavefunction repeats properly after a full cycle

how does this differ from Hψ=Eψ?

particle on a ring = single quantum number m_ℓ ; hamiltonian is KE operator for that dimension

same eigenvalue equation used for particle in a 1D box but wavefunction is constrained by different values

explain this

hamiltonian is sum of KE operators for each direction

wavefunction is the product of 1D wavefunctions

energy is the sum of the energies due to the motion along each direction

explain this

two dimensions and two quantum numbers - ℓ and m_ℓ

Y term is the wavefunction and separates into the Θ term as shown

• separation works differently because of the geometry of the sphere. the wavefunctions differ in more than just their variables

• energy depends on only one of the quantum numbers (ℓ)

what is the wavefunction restricted by?

where the particle can go and how it can move

what