QUANTUM MECHANICS

What is the purpose of quantum mechanics in chemistry?

Theory that lets chemists see and predict the invisible world of electrons

Explains to us how energy is quantisized into levels.

What is a wavefunction in quantum mechanics?

A mathematical function that contains all the dynamical information about a system

• describes probability of finding a particle at position x, doesn’t directly gibe you the probability ( that’s the probability density’s)

What is the Schrödinger equation ?

A second order differential equation used to calculate the wavefunction of a system

4 Requirements of a wave function

1. Must be single-valued

2. Continuous

3. Not infinite over a finite region of space

4. Have a continuos slow(except in special cases)

What is a node ?

A point where wave function passes 0

What does the Born interpretation assume?

The probability density at a point, x, is proportional to the square if the wave function at that point, x

How to normalise a wavefunction and why is it needed ?

When integral over all space of its square modulus is equal to 1

It is required so that wavefunction corresponds to a real physical state, so that probability equals 1and to allow consistency between different states

What is an observable

Measurable property such as bond length, dipole moment

What is an operator

Represents observations and yield information from them

What is the Schrödinger equation?

Essentially in the form of an eiegnvalue equation, specifically for the Hamiltonian operator, the energy operator.

They describe to us how this operator acts on the wavefunction and how it changes over time as well as the discreet energy levels

What is the Hamiltonian operator and its general equation?

Total energy operator

H= T + V ( all are operators)

T=\frac12mv^2 For motion I’m x direction, \frac{h^2}{2m}\frac{d^2\left(x\right)}{\differentialD x^2}

What does a particle in a box allow us to understand?

Motion type: Linear (1D)

Used to understand motion of a particle of gas in 1D container

Can estimate electron excitation energy

Key quantisation: n

What is the Hamilton equation for a particle in a box, and what are the boundaries?

Potential energy can be ignored

-\frac{h^2}{2m}\frac{d^2\psi\left(x\right)}{\differentialD x^2}=E\psi x)

Boundaries are 0<x<L

What does Euler’s relation imply?

e^{\pm ix}=\cos x\pm i\sin x

Solutions for particle in a box and its derivation to final wavefunction form?

\psi_{k}\left(x\right)=Ae^{ikx}+Be^{-ikx}_{^{}}

After derivation

\psi_{k}\left(x\right)=D\sin x

Wavefunction and energy for particle in a box?

\psi_{n}\left(x\right)=\sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}

For 0\le x\le L

E_{n}=\frac{n^2h^2}{8mL}

When n=1,2,3...

remember quantum number n cannot equal 0

What is the irremovable energy ?

Zero point energy = irremovable energy

Particle is never stationery, hence lowest energy level will never be 0

What is the normalisation condition?

The integral of the modules of the wavefunction mover all space must be equal to 1

This way it makes the observable a real and physical value

Steps for normalisation ?

1. Find the wavefunction

2. Write as normalisation condition

3. Solve to find constant

4. Apply boundary or limits

5. Sub constant back in

What does the complex conjugate used for ?

During normalisation, to find a real probability, it makes an imaginary wave function real

What do the derivative, second derivative, partial derivate and integral of a wavefunction tell us?

Wavefunction is the fundamental to all matter as it contains info about particles

But it isn’t measurable e

First derivative - how something(eg WF) changes ( in QM first derivative of WF is related to momentum)

Second derivative - rate of change (curvature of WF, links to KE, high curvature = high KE)

Partial derivations - for when WF depends on more than one variable, will tell you how WF changes due to one variable changing whilst others are kept constant

Integral - total or average value of a property being measure

What does a particle moving on a ring allow us to understand?

Motion is rotational, not linear (2D)

Leading to understanding quantised angular momentum (rotational quantization)

Key quantisation: m

What is the Hamilton equation for a particle on a ring, and what are the boundaries?

H=\frac{h^2}{2I}\frac{d^2}{d\phi^2}=E\psi

0<x<2\pi

What are polar co ordinates used for a particle on a ring

\left(x,y\right)\rightarrow\left(r,\theta\right)

What is inertia and equation?

A measure of an object’s resistance to rotation changes in direction about z axis

I=mr^2

What is the solution wavefunction for a particle on a ring

\psi\left(\phi\right)=Ne^{\pm im_{l}\phi}

Expand to become

\psi\left(\phi\right)=N\left(\cos m_{l}\phi\pm i\sin m_{l}\phi\right)

Wavefunction and energy for particle on a ring ?

\psi\left(\phi\right)=\frac{1}{\sqrt{2\pi}}e^{\pm m_{l}\phi}

When m_{l}=0,\pm1,\pm2....

E_{m_{l}}=\frac{ml^2h^2}{2I}

What do m_{l}=0 and \pm m_{l} Indicate respectively to a particle on a ring

0 means no rotation, so stationary in that sense

+sign means clockwise rotation

-sign means anti-clockwise rotation

What occurs to a particle on a rings motion, as ml increases?

Uses cosine or sine (90 degrees rotated) pathway , but more nodes

What does a particle moving freely in sphere (rigid rotor) allow us to understand?

Motion type: Rotational (3D)

Describing rotation motion of molecules, eg two atoms in a diatomic molecule rotat8ng around their centre of mass

Gives microwaves spectra

Key quantisation: J

What is the Hamilton equation for a particle on a sphere, and what are the boundaries?

E=\frac{-h^2}{2I}\Lambda^2\psi\left(\theta,\phi\right)=E\psi\left(\theta,\phi\right)

This wavefunction comprises of spherical harmonics

Boundary are separate,

As \Theta is a function of variable \theta (north to south) so 0<x<\pi

As \Phi is a function of variable \phi (rotation around z axis) so 0<x<2\pi

What are the spherical polar co ordinates used for a particle in a sphere?

\left(x,y,z\right)\rightarrow\left(r,\theta,\phi\right)

What is the solution wavefunction for a particle on a sphere

Spherical harmonics allows energy to be independent of ml And for every L value there’s (2L+1) degenerate energies

Hence H=\frac{h^2}{2I}L\left(L+1\right)=E

What is the reduced mass and its equations?

Makes diatomic molecules ( both particles which are in motion connected) become a particle of mass \mu

\mu=\frac{m_{A}m_{B}}{m_{A}+m_{B}}

In rotational spectroscopy, what does the quantum number J represent?

How fast is a molecule rotating

Quantises angular momentum of particle due to rotation in steps

Wavefunction and energy for particle for rigid rotor

E=\frac{h^2}{2I}J\left(J+1\right)

Where J=1,2,3...

and I=\mu r_{AB}^{^2}

What does the hamornic oscillator theory allow us to understand?

Motion type: vibrational

Zooms into a particle oscillating about a central point - like two atoms vibrating in a bond

Explains vibrations of molecules giving IR spectra

Shows that vibrations energy is quantised but equally solves

Key quantisation: N

What is the restoring force and equation ?

Hooke’s Law

F=-kx

Meaning negative gradient of potential energy , the restoring force opposes displacement ( eg a string)

Can integrate to find the amount of potential energy from displacement (x=0) (parabola shape/ harmonic potential wall area)

What is the Hamilton equation for a Harmonic oscillator, and what are the boundaries

H=-\frac{h^2}{2m}\frac{d^2}{\differentialD x^2}+\frac12kx^2=E\psi

Wavefunction and energy for a harmonic oscillator system?

\psi_{v}=A_{v}H_{v}\left(\sqrt{2\alpha x}\right)e^{-ax^2}

E_{V}=\frac{h^2}{2}\omega

Where \omega=\sqrt{\frac{k}{n}}

Energy for morse potential system ?

E=\left(v+\frac12\right)h\omega_{e}-\left(v+\frac12\right)^2h\omega_{e}\chi_{e}

Where \chi_{e}=\frac{a^2h}{2\mu\omega} and \omega=\sqrt{\frac{k}{\mu}}

How do the morse potential differ to Harmonic Oscilattor system?

Harmonic oscillator describes bond behaving like a perfect string - symmetric, only valid at equilibrium bond length

Morse potential shows the potential energy of the bond as it stretches ( bond dissociation ) and compresses away from equilibrium bond length

Steps for solving a wave function in quantum mechanics

1. Identify operator required depending on what we need

2. Write H hat and any other variable

3. TISE - for eigenstates and TDSE - for dynamic

4. Soleva and apply boundary condition

5. Normalise and interpret

6. Check limits/sanity check