Quantum mechanics

Question 1

Give the Schrödinger equation in its most basic form

Answer 1

H is the Hamiltonian operator

Question 2

Give the Schrödinger equation for a particle of mass m moving in one direction with energy E

Answer 2

Question 3

State some phenomena that are explained by quantum mechanics

Answer 3

• Photoelectron effect
• Atomic spectra

Question 4

Give an equation for kinetic energy involving momentum

Answer 4

T = p²/2m

Question 5

State the velocity of a free particle

Answer 5

Velocity is constant for a free particle

Question 6

Give the equation for wavevector

Answer 6

Question 7

Give the equation for angular momentum

Answer 7

Question 8

Give the equation for wave velocity

Answer 8

Question 9

Draw out the first 4 waves if a particle is confined to a box

Answer 9

Question 10

State the position of nodes for a standing wave

Answer 10

Question 11

Give and describe Wien’s displacement law

Answer 11

This law gives the observed shifts in the maximum of emission curves

Question 12

Give and describe the Stephan-Boltzmann law

Answer 12

M is the power emittance, ie rate of energy output

Question 13

Give and describe the Rayleigh-Jeans law

Answer 13

Question 14

Describe the photoelectric effect and how it is an example of wave-particle duality

Answer 14

• When a metal is exposed to UV light, electrons are ejected from the metal’s surface
• Below a threshold frequency, no electrons are emitted
• Even at low intensities, electrons are emitted as long as threshold frequency is reached
• The kinetic energy of the electrons varies with frequency but not with intensity

Question 15

State the de Broglie relationship

Answer 15

Question 16

Define the Hamiltonian operator

Answer 16

H = T + V

Hamiltonian operator = kinetic emergy operator + potential energy operator

Question 17

Describe quantum tunnelling

Answer 17

Quantum tunnelling describes the phenomena of a subatomic particle being able to travel through a classically forbidden region. The particle’s behaviour can be explained by it having wavelike properties

V is the potential barrier, E is the energy of the particle, a is the thickness of the barrier in m

Question 18

Describe Heisenberg’s uncertainty principle

Answer 18

“It is impossible to specify simultaneously with arbitrary precision, both the momentum and the position of a particle”

Question 19

Describe cases in which quantum tunnelling is present in chemistry

Answer 19

• The oxidation and reduction of metals. _{^{When ligands are present, the transferred electrons seem to pass through them}}
• Electrical contacts. ^{_{Metal surfaces are usually coated with a thin layer of oxide. Conduction can take place by tunnelling through this layer.}}
• Electrode reactions.
• Photon transfer
• Bond rotation

Question 20

Describe the scanning tunnelling microscope

Answer 20

The microscope consists of an atomically sharp tip. A voltage is applied between the tip and the surface, ensuring they are very close but not touching so that tunnelling occurs

Question 21

Give the momentum operator in 1 dimension

Answer 21

Question 22

Give the Schrödinger equation for a free particle moving in 1 dimension

Answer 22

Question 23

Give examples of a free particle

Answer 23

• A gas molecule in a big container
• An electron conducting in a metal
• An electron beam

Question 24

Give the Schrödinger equation for a particle in a box

Answer 24

The particle is no longer free, although V inside the box is 0. Outside V is infinite - the particle cannot exist at or outside the walls as infinite energy is impossible.

Note: There are boundary conditions. Ψ at the edges of the box = 0

Question 25

State the condition for a solution to the Schrödinger equation to satisfy

Answer 25

operator x function = constant x function

Question 26

Describe the Born interpretation for Ψ

Answer 26

For a particle moving along x, the probability of finding a particle in a infinitesimally small region of space between x and x+dx is proportional to Ψ^2.

The probability therefore must = 1

Question 27

State what |Ψ|² =

Answer 27

|Ψ|² = Ψ*Ψ

Question 28

State the properties of a well behaved wavefunction

Answer 28

• Single valued, as a particle can only have one probability at a certain point
• Finite
• Continuous
• Not equal to 0 everywhere

Question 29

Give the Schrödinger equation for a simple harmonic oscillator

Answer 29

Potential energy has a value.

k is the spring constant

m is the reduced mass of the system

Question 30

Describe the Schrödinger equation for a particle on a ring

Answer 30

We must consider angular momentum.

The boundary condition is that the number of wavelengths must be an integer, otherwise, there will be interference

V = 0 and r, the radius, is constant