QM 1 - translational, vibrational, rotational motion

explain the difference between free translation and confined translation

free = constant speed, no interactions, any energy (not quantised)

confined = energy is restricted to certain values, interactions with wall

for free translational motion - what is potential and kinetic energy?

no interactions = nothing to slow it down, speed it up or change direction of motion

no potential energy, energy is purely kinetic

what is the Schrodinger equation for free translational motion?2

can be written as linear combination or as sine and cosine

what does this mean?

there is no restriction, k can be any non negative real number

all energies are allowed -free translational motion is not quantised

explain what it means if k = 0, and if k>0

for free translational motion, show energy vs momentum graph

explain what graph shows

if k = 0 the particle is still

if k > 0 , the particle is moving and energy equals as shown

• graph shows energy and momentum vary smoothly and continuously = shows no restrictions on k

what are the conditions of the box for particle in a box?

potential energy is 0 inside the box

box length is L

walls are infinitely repulsive (potential energy suddenly shoots up to ∞). this means no transfer of E to the wall

• the wall is infinitely hard = no bounce back/deformation

what are the values of these operators?

same as in free translation

• purely kinetic

what does infinite repulsion mean about boundary conditions for particle in a box?

at the walls (x=0 and x=L), the wavefunction must equal 0

• particle cannot be found there

• this means that |ψ|² = 0 and ψ = 0

what is the consequence of these boundary conditions?

not every wavelength fits properly between the walls

• must be a sine function that starts at 0, oscillates and then returns to 0 at x=L

• must be a whole number of half-wavelengths

this leads to quantisation

what is n? what is N?

n is the quantum number (can only take positive integer values)

• the value of n is the same as the number of half wavelengths the wavefunction has

N is the normalisation constant

show diagram of particle in a box wavefunctions for each quantum number up to n = 4

explain why cos term is not involved in particle in a box and how this is achieved

ψ=0 when x =0 but cos(0)=1

M = 0 to maintain this

explain how the red and green wavelengths are incorrect

the wavelength of red is too long as it hasnt completed enough oscillations by the time it gets to x=L

• there is a discontinuity at x=L as it doesnt hit zero

the wavelength of green is too short, it oscillates too quickly and doesnt land on 0 when x=L

• this leads to a discontinuity

wavelengths must be continuous

explain how the wavelengths not being continuous leads to quantisation of energy

only some wavelengths being acceptable

since wavelength is connected to momentum through the de Broglie equation (p=h/λ)

momentum is connected to KE (K=p²/2m)

• restricting wavelengths means restricting energy

what is the ground state for particle in a box

n =1

no nodes, single arch - 1 half wavelength

what happens as n increases? 3

number of nodes (n-1) increases

curvature of ψ increases, λ decreases, K increases

correspondence principle

why does KE increase as n increases in particle in a box?

the more curved the function is the more KE

the components are moving faster

what is the correspondence principle for particle in a box?

non uniformity of |ψ|² becomes harder to detect

• as you go macroscopic, there is constant probability of finding the particle anywhere in the box

quantum mechanics resembles classical - spends equal time everywhere

what is the zero point energy of particle in a box?

the lowest energy

• confined particle can never stay still

why is the zero point energy not zero?

explain in terms of uncertainty and wavefunction shape

the uncertainty principle says that as the uncertainty of x is finite (confined to region of finite size), the uncertainty of p cannot be zero. if momentum isnt precisely zero, KE cant be zero either

the wavefunction must be zero at the walls but it cannot be zero everywhere (zero probability of finding the particle in the box)

• wavefunction must be curved which means KE

how do the energy gaps change as you go up in particle in a box? why?

energy gaps increase with n

gap between n and n+1 is proportional to 2n+1

• the potential rises more steeply than the potential of a harmonic oscillator (which has constant gaps)

what is the energy gap and ZPE of particle in a box proportional to?

what about for macroscopic?

h²/mL² (energy scale)

• this gives energies that matter for electron in box the size of a chemical bond

for macroscopic, it is too small to be observed

• dont notice quantisation when looking at macroscopic

explain what this shows

• electron in 100 pm box vs marble in 10 cm box

for electron, the ZPE is a few million joules per mole. the energy gap between 2 and 1 is 3x that - these are chemically significant

for the marble, the ZPE is much smaller (of the order of 10^-40 Jmol^-1) and undetectably small. the energy gap is also essentially zero. energies so small cannot be detected

• quantisation is there in principle but for macroscopic, the ELs are so close together that they form what looks like a continuous spectrum

= correspondence principle

when is separation of variables applicable?

when the Hamiltonian is a sum of independent terms

• coordinates that appear in one term do not appear in any other

what is the wavefunction for a particle in a two or three dimensional box?

product of wavefunctions for a particle in a one dimensional box

what is the energy of a particle in a two or three dimensional box?

sum of the energies for each one dimensional component

for a particle in a 1D box - what is the Hamiltonian? what does the energy depend on? and what is the wavefunction?

hamiltonian is the KE operator (second derivative wrt to x)

energy depends on a single quantum number n

wavefunction is a sin function with x as its variable

for a particle in a 2D box - what is the Hamiltonian? is the separation of variables applicable? what is the energy? what is the wavefunction? how many quantum numbers?

hamiltonian has two KE terms - one involving the second derivative wrt x and the other involving the second derivative wrt y

separation of variables is applicable

• the two terms are independent and dont interfere with each other

• 2D can be split into a pair of 1D problems

since separation of variables is applicable, the total E is a sum of two independent 1D energies

wavefunction is a product of two independent 1D wavefunctions

there is now two quantum numbers, one for each dimension

show the Schrödinger equation for 3 dimensions

show particle in a 1D box wavefunctions (n=1 and n=2 overlayed) and explain positive and negative lobes

ground state has a single arch and is positive everywhere in the box

the wavefunction of the first excited state has a node in the middle and has one positive and one negative lobe

show particle in a 2D box wavefunctions for n_x=1 or 2 and n_y=1 or 2

explain the lobes

for ground state (n_x=n_y=1) there is a single positive region

n_x=2 and n_y=1 there is a quantum of excitation in x, there is a vertical nodal line at x=L_x/2

n_x=1 and n_y=2 there is a quantum of excitation in the y direction so there is a horizontal nodal line at y=L_y/2

for n_x=n_y=2 there are nodes in both directions which creates a pattern of 4 lobes with alternating signs

for a square box where L_x=L_y=L, what is the energy formula?

show whether (n_x,n_y)=(3,3) is degenerate or not

show diagram

there is only one state where you can get 18

show whether (n_x,n_y)=(1,2) and (n_x,n_y)=(2,1) are degenerate or not

show diagram

n_x²+n_y²=5 in both cases

there is symmetry in the box so if you rotate by 90 degrees, you swap the x and y directions and the wavefunctions transform into each other

• the wavefunctions have the same energy if excitation along x or y

give an example of accidental degeneracy (for example use n_x²+n_y²=50)

show diagram

(7,1)/(1,7) and (5,5) are not related to each other by symmetry but they are degenerate as they have the same energy