Quantum Mechanics

Question 1

What is quantum mechanics?

Answer 1

Molecular energies or the orientations of nuclear spins can only take discrete values
It predicts that the energy of a particle, such as an electron, can only take a discrete set of values, often called energy levels.
The position of a particle cannot be precisely predicted, only the probability that a particle has a given position can be calculated

Question 2

What is classical mechanics

Answer 2

Predicts that the energy of an object can take any value
Energy varies continuously.
Used to predict the position of an object at any given time

Question 3

What is a main axium (starting position) of quantum mechanics

Answer 3

In quantum mechanics every particle has a wavefunction (Ψ)

Question 4

What does wavefunction tell us?

Answer 4

• Contains all the information about the particle. If Ψ is known we can calculate the energy of the particle
• Is a mathematic function of variables which describes the position of the particle, x, y, z
• Gives a recipe for calculating the probability that the particle is in a small volume dV at a particular position

Question 5

What is the born interpretation of the wavefunction

Answer 5

│Ψ(x, y, z)│² dV
Probability density is work out through squaring the wavefunction

Question 6

Another axiom of quantum mechanics is the Schrodinger equation
What is it

Answer 6

The Strodinger equation allows us to find both the wavefunction and the energy levels for particles (e.g. electrons) in a given situation

Question 7

Why does the form of the Schrodinger equation for a particle change

Answer 7

For a particular situation depends on the way in which the potential energy of the particle varies with position

Question 8

An exact solution to the Schrodinger equation can only be found in a very few simply cases
Give an example

Answer 8

One example is an atom with a single electron - the hydrogen atom

Question 9

What are atomic orbitals wavefunctions

Answer 9

The name for the wavefunction (Ψ) found by solving the Schrodinger equation for an atom with a single electron

Question 10

What is the potential energy of a hydrogen atom

Answer 10

The negatively-charged electron is attracted to the positively-charged atomic nucleus by an electrostatic interaction
Therefore, the potential energy of the electron is proportional to -1/r
Where r is the seperation between the nucleus and the electron

Question 11

When does potential energy become zero

Answer 11

As the seperation between the nucleus and electron becomes very large, the potential energy become zero

Question 12

The magnitude of the potential energy of the electron is inversely proportionate to…

Answer 12

The seperation (r) between the nucleus and the electron
As seperation decreases, the electrostatic interaction become more favourable, resulting in a decrease in potenetial energy

Question 13

Why would you say the 1s wavefunction has spherical symmetry

Answer 13

The value of the 1s wavefunction varies only with the separation of the electron from the nucleus regardless of direction, and so has spherocal symmetry

Question 14

The wavefunction corresponding to the lowest energy is the 1s orbtial
Ψ₁s (r)= A₁s exp (-r/a₀)
What is A₁s and a₀

Answer 14

A₁s is a normalisation constant which ensures that the probability of finding the electron somewhere is equal to 1 (the electron is certain to be somewhere)
a₀ is a constant called the Bohr radius (52.9 pm)

Question 15

The isosurface of a 1s orbital looks like this
How does wavefunction change across the isosurface

Answer 15

The wavefunction takes the same value at any point on the isosurface
The size of the iso-surface depends on the value of the wavefunction

Question 16

What is another way we can determine the position of an electron

Answer 16

Use the radial distribution function P(r) to calculate the probability of finding the electron in a thin shell of radium (r) and thickness (dr)
This gives the probability of finding the electron at a distance (r) from the nucleus, regardless of direction
P(r)= 4πr²│Ψ(r)│²

Question 17

For the 1s orbital, the maximum probability of finding an electron occurs where

Answer 17

At the Bohr radius, a₀

Question 17

For the 1s orbital, the maximum probability of finding an electron occurs where

Answer 17

At the Bohr radius, a₀

Question 18

For atomic orbitals (AOs) there are three quantum numbers
How do quantum numbers differ across types of atomic orbtials

Answer 18

n = principle quantum number
l = orbital quantum number
ml = magnetic quantum number
Each orbital has a unique set of these quantum numbers

Question 19

In an atomic orbital with only 1 electron (¹H)
The energy of an atomic orbital in the hydrogen atom depends on?
How is this worked out?

Answer 19

Only the principal quantum number (n)
n = difference in wavelength of light from transition between energy levels
RH = 2.18 x 10⁻¹⁸

Question 20

The orbital energy is measured on a scale which takes its zero point as…

Answer 20

The situtation where the electron is infintely seperated from the nucleus (ionisation)
In this situation the energy of the electrostatic interaction within the nucleus is zero

Question 21

if n has a value of 2
What are the possible values of l abd ml
What do these values of ml imply about orbitals present

Answer 21

l = 0, 1
ml = -1, 0, +1
There are shells of degenerate orbitals (they have the same energy as one another) with a common* n *
e.g. in this case there are three 2p orbtials and a 2s orbital
as the l vaue increases, a new shell containing a new types of orbital occurs

Question 22

Why do we write the wavefucntion of a hydrogen atom in polar co-ordinates (r, θ, φ) rather than Cartesian ones (x, y z)

Answer 22

Since the hydrogen atom and the potential are spherically symmetrical it is convenient
In spherical polar coordinates each wavefunction cab be written as a productnof a radial part R, which only depends on radius and an angular part Y which only depends on orientation of φ and θ

Question 23

In general, the radial distribution function depends only on the radial part of the wavefunction
This can be found for non spherically symmetric orbitals using

Answer 23

P(r)= r²[R(r)]²

Question 24

For the spherically symmetrical 1s orbital the angular part = 1, so for a 1s orbtial the wavefunction is equal to

Answer 24

The radial part alone

Question 25

What is the L-shell

Answer 25

L-shell : the second innermost shell of electrons surrounding an atomic nucleus

Question 26

For the Radial Wavefunction for a 2s orbital, where is the radial node

Answer 26

The radial part of the 2s orbital passes through zero when the electron is 2a₀ from the nucleus
The zero crossing is known as a radial node
The radial part of the 2p orbital is always positive, so these orbitals do not have a radial node

Question 27

What does this comparision of the 1s, 2s and 2p corresponding radial distribution functions demonstrate

Answer 27

The L-shell, (2s + 2p) wavefunctions decay more slowly than the 1s wavefunction, and so the 2s and 2p orbitals spread into a larger region of space (more diffuse)

Question 28

Describe the iso-surface of the 2p wavefunctions

Answer 28

Each has a posititve lobe and a negative lobe
The wavefunction passes through zero between the lobes (this is shown by the pz orientation where wavefunction passes through the xy plane)

Question 29

What is the Aufbau Principle

Answer 29

States that the electrons in a multi-electron atom occupy the orbitals in the order 1s, 2s, 2p, 3s, 3p etc

Question 30

What is the Pauli Exclusion Principle

Answer 30

No two electrons in any system can have identical values for all 4 quantum numbers
It restricts the number of electrons within an orbital to 2 (ms = ±1/2)
meaning values for n, l, and ml must be different
Hence the electron must also have opposite spins

Question 31

What is Hund’s Rule

Answer 31

Every orbital in a subshell is singllaryly occupied with one electron before any one orbital is doubly occupied