# Quantum Theory and the Wave Nature of Matter

This is a topic from Higher Physics 1B

## Contents |

## Introduction

Quantum physics arose in response to many of the unsolved problems of classical Newtonian mechanics. Attempts to apply the laws of classical physics on the atomic scale were consistently unreliable. Several important unexplained phenomenon were blackbody radiation (the spectrum of EM radiation from a heated object) and the photoelectric effect (emission of electrons from a metal under a light source) which will be covered in this section. One essential difference between quantum physics and classical physics is the conception of energy in EM waves as quantised into 'packets' of energy. This was given by Einstein's interpretation of Planck's law, **E = hf**
Where E is energy in joules

- h is Planck's constant 6.626068 × 10
^{-34}m^{2}kg/s - f is frequency in hertz

- h is Planck's constant 6.626068 × 10

This idea of quantisation was subsequently applied the entire subatomic world.

## Blackbody Radiation

As discussed in the Higher Physics 1A notes, an object at any temperature emits thermal radiation, the characteristics of which depend on surface area and temperature amongst other factors. The thermal radiation consists of a continuous distribution of wavelengths from all portions of the EM spectrum. The nature of this distribution was not explained by classical physics.

- A
**black body**is an ideal absorber and emitter of radiation *Blackbody radiation*is radiation emitted by a black body- A small hole leading to the inside of a hollow object is often used to approximate a black body as the hole acts as a perfect absorber:

- A

### Stefan's Law for Black Bodies

Recalling Stefan's Law of radiation from Higher Physics 1A:

- P = σAeT
^{4}

- P = σAeT

Where P is power in watts

- σ is Stefan-Boltzmann's constant (5.67 x 10
^{-8}W/(m^{2}K^{4}) - A is surface area in metres squared
- e is emissivity where 0≤ e ≤1
- T is temperature in Kelvin

- σ is Stefan-Boltzmann's constant (5.67 x 10

- It is clear that for a black body e = 1 and so:

- P = σAT
^{4}

- P = σAT

- Intensity can be expressed by dividing power by the area as follows:

**I = P/A = σT**^{4}

### Wien's Displacement Law

Wien's law describes the trend in the distribution of wavelengths in blackbody radiation as follows: The peak of the distribution shifts to shorter wavelengths as the temperature increases. It gives the following equation for the wavelength which is emitted at the highest intensity:

**λ**(metres-Kelvin)_{max}T = 2.898 x 10^{-3}mK

This trend is seen in the following figure, in which intensity curves for black bodies of different temperatures show the peak tending to the left (shorter wavelengths):

*You may be asked to roughly sketch this curve or trend of curves in an exam, or at very least describe it.*

### Rayleigh-Jeans Law

- The Rayleigh-Jeans Law was an early attempt to model blackbody radiation using classical physics:

**I(λ,T) = 2πck**_{b}T / λ^{4}

- The equation was fairly accurate at long wavelengths, however at lower wavelengths it drastically differed from experimental results such that you would have infinite energy as the wavelength tended toward zero. This was known as the
**ultra-violet catastrophe**. - Planck eventually formulated an equation which suited experimental data, and which, at large wavelengths, was the same as the Rayleigh-Jeans Law

## Planck's Law and Energy-Level Diagrams

The aforementioned Planck's law (E = hf) explains blackbody radiation:

- In atoms the electrons can orbit the nucleus in various 'shells' or energy-levels based on how much energy the electron has
- When an object is hot, it radiates EM waves in the form of photons (recall that light is both waves and particles simultaneously)
- This energy is released as an electron drops from a higher energy shell to a lower one
- The specific energy values of different shells results in the quantised energy values of emitted EM waves, as the energy of the wave equals the difference in the specific energy values of the shells
- As seen in the distribution, it is a matter of probability what the energy of the EM wave emitted by a black body will be, and so the average energy is given by weighting the energy by the probability that it is emitted (this probability is proportional to e
^{-E/kBT}where E is the energy)

The following is an energy-level diagram which can be used to model the change in energy-levels of electrons during blackbody radiation

- Horizontal lines reflect allowed energy levels for electrons
- The double-sided arrows reflect allowed transitions

## The Photoelectric Effect

The photoelectric effect is a phenomenon which to some extent is the reverse of blackbody radiation, whereby incident light (EM radiation) upon a metal causes electrons to be emitted from that metal (known as **photoelectrons**). There were four main disagreements between classical predictions and experimental observations:
1. Relationship between light intensity and photoelectron kinetic energy:

- Classical prediction: Electrons absorb energy continually from the electric field of incident light waves, such that as intensity increases photoelectrons should be emitted with higher kinetic energy
- Experimental result: The maximum kinetic energy is independent of intensity

2. Time period between incident light and photoelectron emission:

- Classical prediction: For low light intensities there will be a measurable time interval between light being incident upon the metal and photoelectrons being ejected, as enough energy is absorbed by the electron for it to be ejected
- Experimental result: Photoelectrons are emitted almost instantly, even at low intensities

3. Dependance of photoelectron emission on light intensity:

- Classical prediction: Photoelectrons will be emitted at any frequency given high enough light intensity
- Experimental result: Photoelectrons are only emitted above a
**threshold frequency**, regardless of intensity

4. Relationship between light frequency and photoelectron kinetic energy:

- Classical prediction: There should be no correlation between frequency and kinetic energy of photoelectrons
- Experimental result: Photoelectron kinetic energy is proportional to light frequency

Einstein's explanation for the photoelectric effect which explains these four disagreements is as follows:

- As previously mentioned, electrons orbit the nucleus of an atom in various 'shells' or energy-levels based on how much energy the electron has
- When a photon from incident light strikes a metallic surface the energy of that photon (given by Planck's law, E = hf) is transferred to an electron within the metal such that it might jump from a lower energy level to a higher one, or be freed entirely from the atom
- Because photon energy is proportional to frequency, there exists a
**threshold frequency**below which no photoelectrons will be emitted because not enough energy is given to free the electron. This explains why light of any colour does not necessarily cause the photoelectric effect - As a result of Planck's law, the energy of photoelectrons is not dependant upon the intensity of incident light as classical physics suggests. Intensity only affects the number of photoelectrons emitted in a given time period, and so intensity is proportional to current in photoelectric circuits
- Photoelectrons are emitted almost instantaneously if they are to be emitted at all, as the build up of energy described by classical predictions is impossible below the threshold frequency

### The Work Function for the Photoelectric Effect

- When a photoelectron is emitted from a metal its kinetic energy is provided by the energy of the incident light (E = hf)
- Some of this energy goes into work to free the electron from the nucleus to which is it attracted, This energy is denoted the symbol
**φ**and called the**work function**of the metal, which varies for different metals - Thus the maximum kinetic energy of the photoelectron after doing work to escape the nucleus is given by:

**K**_{max}= hf - φ

### The Threshold Frequency

- The cutoff point at which a photoelectron has enough energy to escape the metal is given by:

- K
_{max}= 0 - hf - φ = 0
**f**_{c}= φ/h

- K

- Thus the maximum wavelength for the photoelectric effect to occur is given by:

**λ**_{c}= c / f_{c}= hc / φ

## The Particle-Wave Duality, De Broglie's Equation and the Bohr Atom

As previously established there exists a duality whereby matter is simultaneously comprised of waves and particles such that all particles are waves and all waves are particles. The decision to model a phenomenon using particles or waves is based on what is practical for the given situation.

- The
**principle of complementarity**states that wave and particle models for matter or radiation complement each other and that neither model can be used exclusively to adequately explain phenomenon. The types of entities that clearly exhibit both particle and wave properities is called a**quantum particle** - An
**ideal particle**has zero size, such that its mass is localised at a point in space, while an**ideal wave**has a single frequency and is infinitely long such that it is unlocalised in space, its mass so spread out that it appears to have none - De Broglie gave the wavelength as any particle as a function of its momentum:

**λ = h / mv**

- The

Where λ is the wavelength in metres

- h is Planck's constant
- m is mass
- v is velocity (for a photon the value of v is always the speed of light in the given medium)

- By analogy with photons De Broglie suggested the frequency of any particle to be given as follows:

**f = E / h**

- This particle-wave duality applied to all matter was confirmed by the Davisson-Germer experiment in which electrons where shown to exhibit diffraction effects from which their wavelength was measured
- Bohr rearranged this equation to show the orbital radii of electrons around the atom, by noting that an integer number of wavelengths must equal an integer number of orbital circumferences:

- nλ = n2πr and so
**mvr = nh / 2π**

## Wave Packets

A way of reconciling particles and waves is to consider multiple waves superimposed such that they add constructively at one point and destructively everywhere else, such that this small region, called a **wave packet** contains all the momentum of the wave, acting like a particle.
This concept gives rise to the following equations:

**Phase speed**of a wave in a wave packet is**v**_{phase}= w / k

(where w is the angular frequency and k is the angular wave number)

**Group speed**of the wave packet itself is**v**_{group}= dw / dk

(the rate of change of angular frequency with respect to the angular wave number)

## The Uncertainty Principle

The uncertainty principle, formulated by Werner Heisenberg, states that it is impossible to make simultaneous measurements of a particle's position and momentum with infinite accuracy. this is contrary to classical physics.

*One way to understand this claim is to consider an electron. In order to measure its position it needs to be 'seen' by some form of measuring device. This means that another particle (another electron or photon) is bounced off the particle such that its position can be determined by equipment (consider the analogy that our eyes see and determine position by receiving photons reflected off surfaces). This collision will slightly change the particles momentum, such that the momentum cannot be measured simultaneously to position with complete accuracy. The converse is true, because measuring a particle's momentum through a collision changes its position. Essentially the act of measuring one property slightly alters the other*

The minimum amount of uncertainty is given as follows:

**ΔxΔp ≥ ħ / 2**also**ΔEΔt ≥ ħ / 2**

Where ħ is equal to h / 2π

The latter of these implies a temporary violation of the law of conservation of energy for time Δt