Category: 18 Quantum Physics

# Heisenberg Uncertainty Principle

There is no lack of videos explaining the HUP on youTube. Here are a few good ones.

The HUP does not arise from measurement. It is inherent in the wave-particle nature of wave-particles.

The HUP is embedded in the single slit diffraction pattern. Who would have thought of that?

# Emission Spectrum

If you view the light of discharge tubes through a diffraction grating, you can see the emission spectrum directly.

To make sure you know what a non-discrete spectrum look like, this video shows you (1) the continuous spectrum of a filament lamp, (2) the discrete line spectrum of a LED and (3) the “combo” spectrum of a fluorescent lamp.

# What is a Photon?

A photon is a packet of electromagnetic radiation energy. The amount of energy of a photon is related to the frequency (and wavelength) of the EM radiation by

$E=hf=\frac{hc}{\lambda }$

where h is the Planck’s constant.

For example, monochromatic red light of wavelength 750 nm consists of a stream of photons (travelling at light speed) each carrying 1.7 eV of energy,

$E=\frac{hc}{\lambda }=\frac{(6.63\times {{10}^{-34}})(3.00\times {{10}^{8}})}{750\times {{10}^{-9}}}=2.65\times {{10}^{-19}}\text{ J}=1.7\text{ eV}$

whereas violet light of wavelength 400 nm delivers energy in discrete packets each carrying 3.1 eV of energy.

$E=\frac{hc}{\lambda }=\frac{(6.63\times {{10}^{-34}})(3.00\times {{10}^{8}})}{400\times {{10}^{-9}}}=4.97\times {{10}^{-19}}\text{ J}=3.1\text{ eV}$

# Photon Model of Light

According to the photon model of light, electromagnetic radiation energy is delivered in discrete packets called photons.

Between two monochromatic lights of the same wavelength but different power, the one with the higher power is delivering more photons per unit time.

Between two monochromatic lights of different wavelength but the same power, the one with the shorter wavelength delivers fewer photons per unit time, but each photon packs a larger amount of energy.

# The Photoelectric Equation

In the wave model, a light wave delivers continuous energy which is shared by all the electrons in the metal. In the photon model, however, an individual photon deliver discrete energy to an individual electron. This results in a simple energy equation

$KE=hf-WD$

Meaning the KE of the liberated photoelectron is equal to the energy supplied by a single photon  minus the required work done to break free from the metal lattice.

The photoelectrons which are liberated usually reside in the valence band. Even among the valence electrons, some of them are bound more tightly to the metallic lattice than others and thus must do more work before they can escape from the metal lattice. This explains why liberated photoelectrons have a range of kinetic energy even when monochromatic light is used (illustrated in the energy diagram below).

The least tightly bound valence electrons (those occupying the topmost energy levels in the valence band) are the ones which requires the least amount of work before breaking free from the metal lattice. This minimum work is called the work function Φ. The work function is different for different metal. For example, the work function for sodium, aluminium and gold are about 2.4 eV, 4.1 eV and 5.1 eV respectively.

The maximum KE of the liberated photoelectron is thus related to the photon energy and the work function as follow:

$K{{E}_{\max }}=hf-\Phi$

This equation highlights the fact that in the photon model, the photoelectric effect is a one-photon-one-electron interaction: an individual electron absorbs the energy of an individual photon to overcome the work function of the metal. We are going to see how this model can explain all the experimental observations that the wave model cannot.

## Why KEmax is dependent on Frequency but not Intensity of light?

Increasing the intensity of light merely increases the number of photons arriving at the metal surface per unit area per unit time. The rate of emission of photoelectrons will increase, resulting in a higher saturation current. But the energy delivered by each individual photon to each individual electron remains unchanged. So the maximum KE however remains unchanged.

On the other hand, increasing the frequency of light used means that light energy is delivered in larger quantum. With each individual photon delivering a larger quantum of energy, an individual electron absorbs a higher quantum of energy and escapes with higher KEmax. Remember that $K{{E}_{\max }}=hf-\Phi$.

The higher KEmax results in a higher stopping potential Vs since

$K{{E}_{\max }}=e{{V}_{s}}$

# Why there is a Threshold Frequency?

The frequency at which the energy of a photon matches the work function of a metal is called the threshold frequency f0.

$h{{f}_{0}}=\Phi$

Below the threshold frequency, the energy of each individual photon in a light beam will be lower than the work function, meaning each photon does not supply sufficient energy for a single electron can be liberated. (It is not possible for an electron absorb multiple photons because the chance of an electron being hit by two photons simultaneously at the same time is practically zero.) Increasing the intensity of the light does not help at all, since this only increases the number of photons per unit time, but the energy of each individual photon is unchanged.

# Why Zero Time Delay?

Unlike the wave model which requires the energy delivered to be shared by all the electrons, the photon model allows an electron to be liberated the moment it absorbs the energy of one photon. This explains the experimental observation that a photoelectric current (albeit a small one) is registered the moment the light is switched on.

# Intensity and frequency of light as a wave

According to the wave model of light, the difference between a bright and dim light is in the amplitude of the wave.

And the difference between light of different color is in the wavelength (or frequency).

# The Wave model’s predictions

If light is a wave, a beam of light illuminating the metal surface would be delivering energy to trillions of electrons simultaneously at the same time. The electrons would have to share and accumulate the energy over time, before they have enough energy to be liberated from the metal. So there should be a time lag between illumination and emission.

Remember that a wave’s power is proportional to the amplitude-square of the wave. So whatever the frequency of light, we should be able to increase the rate of supply of energy to the electrons by increasing the intensity of the light.

So regardless of the frequency of light used, as we increase the intensity of light, we expect (1) more photoelectrons to be emitted per unit time (higher rate of emission), (2) more energetic photoelectrons (higher KEmax) and (3) shorter time lag between illumination and emission.

# Actual experimental observations

Experimental results however produce 3 contradictions.

(1) Increasing the intensity of light increases the saturation current but not the stopping potential. The stopping potential increases only when higher frequency light is used. (The wave model predicts that KEmax should increase with intensity of light).

(2) If the frequency of light used is below a certain threshold frequency, the photoelectric current will still remain at zero regardless of the intensity of the light. (The wave model predicts that photoelectric effect should occur as long as the intensity is high enough, regardless of frequency)

(3) The microammeter registers a (small) current immediately after the light is switched on, even when very low intensity light is used. (The wave model predicts that there should be noticeable time at low intensity when the rate of supply of energy is low)

## Three observations that show light is not a wave

In summary, the following are the three experimental observations that cast doubt on the wave model of light. Eventually, these three observations will provide evidence for the particulate nature of light.

• KEmax is dependent on the frequency, but indepenent of intensity of light.
• Photoelectric effect does not occur below the threshold frequency, regardless of intensity.
• There is zero time lag even at low intensity.

# What is Photoelectric Effect?

Electrons can be liberated from a cool metal surface when light (electromagnetic radiation) of sufficiently high frequency is incident upon it. This phenomenon is known as the photoelectric effect.

# What are Photoelectrons?

Conduction electrons in the metal are bound to the lattice. The light beam provides the required energy for the electrons to break free from the metalic bonds. The liberated electrons are called photoelectrons.

## Rate and emission and KE of photoelectrons

Two important quantities provide clues for how light energy is delivered: (1) the rate of emission of photoelectrons, which is the number of photoelectrons emitted per unit time, and (2) the kinetic energy of the photoelectrons.

It’s important to grasp the difference between “rate” and “speed”. For example, the photoelectrons on the left are emitted at higher rate, but at lower speed compared to those on the right.

# Photoelectric Current

The experimental set-up to study the photoelectric effect consists of the emitter plate (from which photoelectrons are emitted) and a collector plate (at which photoelectrons are collected). When connected with a piece of wire and a microammeter, the microammeter measures a current. This current is called the photoelectric current.

# Saturation Current

Without any bias voltage, only a fraction of the photoelectrons that leave the plates arrive at the collector.

If we connect a battery between the emitter and the collector such that the collector is at a higher potential than the emitter, we establish what’s called a positive bias between the plates. A positive bias produces an electric field between the plates that attracts photoelectrons towards the collect.

Increasing the positive bias thus increases the percentage of photoelectrons collected, resulting in a higher and higher photoelectric current. The maximum is reached when every single photoelectron is collected. The maximum photoelectric current is called the saturation current Isat.

The saturation current can be used to calculate the rate of emission of photoelectrons because

For example, a saturation current of 1.2 uA would imply an emission rate of 7.5 x 1012 photoelectrons per second.

# Stopping Potential

If we connect the battery such that the collector is at a lower potential than the emitter, we establish what’s called a negative bias between the plates. A negative bias produces an electric field between the plates that repels photoelectrons away from the collect.

So photoelectrons must gain electric potential energy at the expense of losing kinetic energy as they travel towards the collector. For example, if there is a negative bias of 3 V, a photoelectron must lose 3 eV of KE in order to gain 3 eV of EPE if it were to reach the collect. This implies that only photoelectrons with initial KE ≥ 3 eV can arrive at the collector. Those with initial KE < 3 eV will be turned back before reaching the collector.

As photoelectrons are emitted with a range of KE, increasing the negative bias results in less and less photoelectrons arriving at the collector, and thus lower and lower photoelectric current, until the current becomes zero when even the most energetic photoelectron does not have sufficient initial KE to overcome the potential energy barrier between the plates. The negative bias at which this occurs is called the stopping potential Vs.

The stopping potential can be used to calculate the maximum KE of photoelectrons because

eVs = KEmax

For example, a stopping potential of 1.3 V would imply a KEmax of 1.3 eV.

# I–V graph

Shown above is a typical IV graph for a photoelectric experiment. It shows the variation of the photoelectric current against the bias voltage. The two important pieces of information provided by an I-V graph are (1) the saturation current, which allows us to calculate the rate of emission of photoelectrons, and (2) the stopping potential Vs which allows us to calculate the KEmax  of photoelections.

Finally, we are ready to examine the many surprises the photoelectric effect has in store for us.