quantum mechanics part 1

what is the wavelength?

the distance between two wave crests or troughs

what is the frequency?

the number of wave crests that pass over the origin every second

what is black body radiation?

the light radiated by an object when it’s hot

what is emitted when light is shone on a metal surface?

what is expected to happen (energy)? what actually happens?

how was this resolved?

electrons

energy of electrons should depend on intensity of incident light. instead, electrons are emitted above a certain frequency of UV

resolved by considering light as photons

what is the equation for kinetic energy of ejected electron?

m is mass

v is speed

Φ is work function of material (energy to remove electron from surface)

how does the atomic spectrum of hydrogen hint at quantisation?

when heat or electrical energy is applied, it emits an atomic spectrum with discrete wavelengths

• only certain frequencies are emitted

• different for each element

emission spectrum

• appearance

• where are transitions to and from?

• what does spectral line look like?

bright lines against dark background

transitions are from higher to lower energy levels

absorption spectrum

• appearance

• where are transitions to and from?

• what does spectral line look like?

dark line against bright background

transitions are from low to high energy levels

what are Paschen series transitions?

Transitions to n=3

what are Balmer series transitions?

transitions to n=2

what are Lyman series transitions?

transitions to n=1

what is the equation for frequency of light in Hz for hydrogen?

R_H is the Rydberg constant (=3.29×10¹⁵ Hz )

n₁, n₂ are integers representing ELs

what is the difference between constructive and destructive interference of waves?

constructive = in phase

destructive = out of phase

how does amplitude change with constructive and destructive interference?

constructive = waves add to give greater amplitude

destructive = cancel out to give reduced/zero amplitude

what was Bohr’s model of the atom?

electrons move around the nucleus in a fixed orbit

why do macroscopic objects not show perceptible wave like properties?

the de Broglie wavelength is too small

diagram of particle moving in circle

show r, m, v and centripetal force

how to calculate angular momentum?

L=mvr

r is radius of orbit

how to calculate centripetal force?

why is there an electrostatic force pulling electron into centre of orbit?

nucleus has protons and is positively charged

attraction between electrons and protons pulling into nucleus

how to calculate electrostatic force?

Q₁ and Q₂ are point charges

ε₀ is constant, permittivity of free space

what does quantisation mean?

there are discrete levels of energies in an atom, rather than continuous spectrum

what happens if wave is out of phase as it travels in orbit?

starts to destructively interfere and you won’t have a wave anymore

what is the condition of constructive interference of electron wave as it travels around the nucleus?

you need exact number of wavelengths going around the circumference so it starts to overlap

2πr = n𝜆 (circumference)

how is this derived?

what is e? what is Z? what is v?

v is velocity of electron

e is charge (-e for electron and +e for protons in nucleus)

Z is number of protons (atomic number)

for an electron in orbit around the nucleus, what is centripetal force equal to?

electrostatic force

using the fact electrostatic and centripetal forces are equal

find equation for r_n

how to find potential energy of electron in orbit?

E = force x distance (r_n)

= electrostatic force x r_n

how to find energy of electron?

kinetic energy + potential energy

use expression for r_n and the energy of electron to find equation for E_n

equation for frequency using rydberg constant

how to modify if calculating other than H?

multiply by Z²

what were the problems with Bohr’s model?

·      He couldn’t explain the quantisation condition for angular momentum. (only certain orbits allowed)

·      Classically, an orbiting charged particle would emit electromagnetic radiation, lose energy and fall into the nucleus.

·      While the model could explain the atomic spectrum of hydrogen, it couldn’t predict the spectra of other atoms.

what is the time independent Schrödinger equation?

what do the differentials in the Schrodinger equation show?

how wavefunction changes with distance from the nucleus

e.g. d²/dx² describes how ψ changes along x axis with constant y and x

what type of problems are solved with the schrodinger equation?

in the case of atoms?

two body problems

this is a nucleus and singular electron in the case of atoms

what is kinetic energy

what is potential energy

kinetic = energy of motion

potential = dependent on position

what is the wavefunction ψ (psi) - what does it contain info about

is it directly measurable?

contains info on the properties and behaviour of a particle but is not directly measurable

what is ψ² proportional to?

what does it mean if ψ² is high?

the probability of finding the particle in a small volume dV

if ψ² is high, the probability of finding the electron is high

what is ψ² for electrons?

electron density

how does schrodinger equation agree with Rydberg and Bohr for hydrogen?

matches predictions that energy depends on principle quantum number

what is the Heisenberg uncertainty principle?

what does l describe

orbital angular quantum number

how many nodes - ie which orbital it is

what are the values of l for s, p, and d orbitals

s: l = 0

p: l = 1

d: l = 2

for n = 1, n = 2, n =3, what are the values of l?

n = 1 : l = 0

n = 2 : l = 0, 1

n = 3: l = 0, 1, 2 (s,p,d)

what does m_l describe

magnetic quantum number

the direction of the orbital

for l = 0, l = 1, l =2

what are the values of m_l?

l = 0 (s orbital), m_l = 0

l = 1 (p orbital), m_l = -1, 0, 1

l = 2 (d orbital), m_l = -2, -1, 0, 1, 2

why would you want to calculate wavefunction as a function of spherical coordinates rather than polar?

atoms are spherical

how to calculate x,y,z using polar coordinates?

x=r sin⁡θ cos⁡ϕ

y=r sin⁡θ sin⁡ϕ

z=r cos⁡θ

what is the radial wavefunction dependent on?

n and l

what does the angular wavefunction depend on?

l and m_l

how many nodes does

s orbital

p orbital

d orbital

f orbital have?

s is n-1

p is n-2

d is n-3

f is n-4

why is the s orbital spherically symmetric?

no angular dependence

which orbitals are not zero at the nucleus (radial nodes)?

s orbitals

what are the radial nodes graph shapes for n =1,2,3?

where is the highest probability for the electron in radial distribution functions?

the maximum value is most probable distance

what is the spin quantum number?

which way is clockwise/anticlockwise?

represents the direction of spin, m_s = ± ½

what is the Aufbau principle?

Lowest energy levels are filled first

what do the energy levels for multi-electron atoms depend on?

• principle quantum number n (which EL)

• orbital angular quantum number l (which orbital)

for hydrogen, what do energy levels depend on?

what does this explain?

energy levels depend only on n

• energy levels are degenerate for different values of l and m_l for given value of n

this is why Bohr model worked for hydrogen

what is Pauli exclusion principle?

no two electrons in same region of space can have the same 4 quantum numbers (n, l, ml, ms)

• this means 2 electrons per orbital (ms = ± 1/2)

what happens to electrons with same quantum numbers when electron orbitals overlap?

electrons in different atoms with same quantum numbers repel

what are the different types of energy stored?

electronic, vibration, rotation, translation

why can the different forms of energy stored in a molecule be treated independently?

electronic transitions are much faster than bond vibrational motion, faster than rotations, faster than translational motion

in particle in a box, what is the potential energy inside the box?

0

in particle in a box, what is the potential energy outside the box?

∞

what is the wavefunction at the edge of the box?

how does this affect quantisation?

0

the particle cannot exist in regions of space where 𝜓 (𝑥) = 0, so confinement leads to quantisation of energy levels

what is the energy inside the box?

all energy is kinetic

how does increasing particle size affect wavelength and ΔE?

increases emission wavelength and decreases gaps between ELs