#### Wave Particle Duality and Quantum Theory

#### Physics (Year 12)

# Photon

As explained in the previous section, light exhibits both wave and particle-like properties. The particle associated with light is called photon. Photons are massless (ie. have 0 mass) and travel at the speed of light, c.

Photons carry discrete packets of energy, and these packets of energy are called quanta. Hence photons are occasionally referred to as light quanta. The energy of the light quanta depends on its frequency (ie. the light’s frequency). The relationship between the energy of light and its frequency is given by:

Since we know that c=λf, then the equation above can be expressed as:

# Blackbody radiation

Evert object which has a temperature above absolute 0 (ie. 0 kelvin), emit and absorb electromagnetic radiation. We will specifically focus on theoretical objects known as blackbodies. A blackbody is considered to an ideal absorber and emitter of radiation. Since it radiates and absorbs at the same rate, it is always in equilibrium with its surroundings and hence has a constant temperature. Below is a graph of a blackbody emission spectrum for temperatures ranging from 3500K to 5500K.

*diagram*

As seen in the graph above, as the temperature increases, the peak wavelength decreases which corresponds to an increase in peak frequency and energy (according to the equations; c=λf and E=hf). This can also be seen in the graph as the peak energy density increasing as temperature increases.

The above graph is a result of a quantum interpretation for observations. This theory was first introduced by Max Planck (hence Planck’s constant). To make his theory agree with the experimental observations, Planck introduced 2 postulates (a postulate is something that is assumed to be true). The 2 postulates are:

Molecules vibrate at discrete energies. These energies are given by E = hf

Molecules emit and absorb radiation in discrete packets of energies known as photons

Before Max Planck introduced his theory, the general accepted theory was based on the Rayleigh-Jean law. However, this classical theory had an issue. Based on the Rayleigh-Jean law, as the wavelength decreases, the energy density increases and approaches infinity. This is impossible since you cannot have infinite energy. This is now known as the ultraviolet catastrophe. It is called that because the energy density approaches infinite in the wavelength range that corresponds to ultraviolet radiation.

*diagram*

# Photoelectric effect

The photoelectric effect is an important phenomena because it is one of the most significant experiments that provides evidence for the particle-like nature of light. A diagram of the effect is below:

*diagram*

When light above a specific frequency hits the piece of metal, an electron is ejected (this ejected electron is called a photoelectron) which travels to the opposite plate. Hence this flow of electrons results in a current (this current is called photocurrent).

The metal which the light shines on is sometimes referred to as the target metal, or the cathode metal. For different pieces of metal, there is a different frequency below which no photoelectrons are ejected. This frequency is known as the threshold frequency, and it varies according to the metal.

Sometimes, a voltage can be applied to the circuit which influences the amount of photoelectrons collected on the anode metal. If the voltage is positive, then the negatively charged photoelectrons will be accelerated to the positively charged anode metal. This ensures that every photoelectrons is collected. If the voltage is negative, then the negatively charged photoelectrons will be repelled away from the positively charged anode metal. This reduces the amount of photoelectrons reaching the anode. The voltage at which no photoelectrons reach the anode is called the stopping voltage. There is a relationship between the stopping voltage and the maximum kinetic energy of emitted photoelectrons. The relationship is:

Photoelectrons have the maximum kinetic energy when they are ejected from the most outer layer of the atom. Other photoelectrons come from the deeper layers of the atom. As they make their way out of the atom they collide with other electrons and hence lose kinetic energy.

The minimum energy required to eject a photoelectron from a metal is known as the work function. If the energy of the incoming light is not greater than the work function, then no photoelectrons will be emitted. Work function is related to the metal’s threshold frequency and since we know that threshold frequency varies, work function also varies. The relationship between work function and threshold frequency is given by:

If the energy of the incoming light is greater than the work function, then the extra energy is transformed into the kinetic energy of the photoelectron. This is described by:

You can notice that the above equation is of the form y=mx+c. So if you were to graph the above equation, the result would be a straight line where the gradient of the line will be Planck’s constant, and the y-intercept will be the work function.

# Benefits of the photoelectric effect

The photoelectric effect cannot be explained by using the wave model of light. According to the wave model, the frequency of the incoming light should not affect if a photoelectron is ejected or not. This is because since a wave is a form of continuous energy transfer, the energy can build up. Implying that even low energy light (ie. light of a lower frequency) can cause photoelectrons to be ejected, if they are left incident for long enough. If this is assumed to be true, there should be a time delay the light striking the metal and photoelectrons being ejected, since the wave needs to ‘build up’. However, this wasn’t observed during experiments.

What was observed is that frequency doesn’t affect the maximum current produced by the photoelectric effect. What does affect the current, is the intensity of light. Higher intensity light striking the metal means more photons are interacting with the atoms and more photoelectrons are being ejected. As current is the flow of electrons, a higher flow of photoelectrons results in a higher current.

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