6. Atomic and Nuclear Physics Chapter 6.5 Quantum theory and the uncertainty principle

When a atomic gas, like hydrogen, is either heated at a high temperature or exposed to a high electric field it will glow and emit light. The emitted light can be analysed using a spectrometer that will split the light into its component wavelengths Atomic spectra The lines produced make the emission spectrum of hydrogen.

Other gases will have similar spectra. But different gases will have emission lines at different wavelengths Atomic spectra

A similar phenomenon takes place when white light is allowed to pass through hydrogen gas. Most light will pass through but a series of black lines are also seen. This constitutes an absorption spectrum, where the black lines correspond to the radiation absorbed by the gas Atomic spectra

Sun’s absorption spectrum

Atomic spectra The striking feature of emission and absorption spectra is the fact that the emission and absorption lines are at specific wavelengths for a particular gas.

In 1885 Johann Balmer, a Swiss physicist, discovered, by trial and error, that the wavelengths in the emission spectrum of hydrogen were given by the formula: Explanation of spectra Johann Balmer ( ) where n may take integer values 3, 4, 5, … and R is a constant number

Since the emitted light from a gas carries energy, it is reasonable to assume that the emitted energy is equal to the difference between the total energy of the atom before and after the emission. Since the emitted light consists of photons of a specific wavelength, it follows that the emitted energy is also of a specific amount since the energy of a photon is given by: Explanation of spectra This means that the energy of the atom is discrete, that is, not continuous.

If the energy of the atom were continuous the emission of light wouldn't always be a set of specific amounts. The first attempt to explain these observations came with the “electron in a box” model. Imagine that an electron is confined in a box of linear size L. If the electron is treated as a wave, it will have a wavelength given by: The “electron in a box” model x=0x=L the electron can only be found somewhere along this line

If the electron behaves as a wave, then:  The wave is zero at the edges of the box  The wave is a standing wave as the electron does not lose energy This means that the wave will have nodes at x=0 and x=L. This implies that the wavelength must be related to the size of the box through: The “electron in a box” model Where n is an integer

Therefore, the momentum of the electron is: The “electron in a box” model The kinetic energy is then:

This result shows that, because the electron was treated as a standing wave in a “box”, it was deduced that the electron’s energy is quantized or discrete: The “electron in a box” model However, this model is not correct but because it shows that energy can be discrete it points the way to the correct answer.

In 1926, the Austrian physicist Erwin Schrödinger provided a realistic quantum model for the behaviour of electrons in atoms. The Schrödinger theory assumes that there is a wave associated to the electron (just like de Bröglie had assumed) This wave is called wavefunction and represented by: The Schrödinger theory Erwin Schrödinger ( ) This wave is a function of position x and time t. Through differentiation, it can be solved to find the Schrödinger function:

The Schrödinger theory This wave is a function of position x and time t. Through differentiation, it can be solved to find the Schrödinger function: where  r (x, y z) is the particle's position in three-dimensional space,  is the wavefunction, which is the amplitude for the particle to have a given position r at any given time t.  m is the mass of the particle.  V (r) is the potential energy of the particle at each position r.

The Schrödinger theory The German physicist Max Born interpreted Schrödinger's equation and suggested that: can be used to find the probability of finding an electron near position x at time t. This means that the equation cannot tell exactly where to find the electron. This notion represented a radical change from classical physics, where objects had well-defined positions.

The Schrödinger theory When Schrödinger's equation is applied to the electron in a hydrogen atom, it gives results similar to those found using the “electron in a box” model. It predicts that the total energy of the electron (E k + E p ) is given by: where n is an integer and C a constant equal to: where: - k is the constant in Coulomb’s law - m is the mass of the electron - e is the charge of the electron and - h is Planck’s constant

The Schrödinger theory Substituting the constants by their values, it is found that: In other words, this theory predicts that the electron in the hydrogen atom has quantized energy. The model also predicts that if the electron is at a high energy level, it can make a transition to a lower level. In that process it emits a photon of energy equal to the difference in energy between the levels of the transition. So,

The Schrödinger theory Because the energy of the photon is given by E = hf, knowing the energy level difference, we can calculate the frequency and wavelength of the emitted photon. Furthermore, the theory also predicts the probability that a particular transition will occur. energy This is essential to understand why some spectral lines are brighter than others. Thus, the Schrödinger theory explains atomic spectra. 0 eV eV n=1 n=2 n=3 n=4 n=5 high n energy levels very close to each other

The Schrödinger theory The graph below shows the variation of the probability distribution function with distance r from the nucleus for the energy level n=1 of the hydrogen atom. The shaded area is the probability for finding the electron at a distance from the nucleus between r = a and r = b. Probability density ψ 2 r (x m) ab

The Heisenberg Uncertainty Principle Werner Heisenberg, a German physicist and one of the founders of Quantum Mechanics, discovered the principle in Heisenberg said that if the electron behaves simultaneously as a wave and as a particle, we cannot divide physical objects as either particles or waves. Applied to position and momentum, Heisenberg uncertainty principle states that: Werner Heisenberg ( ) It is NOT possible to measure simultaneously the position and momentum of something with indefinite precision.

The Heisenberg Uncertainty Principle This has nothing to do with imperfect measuring devices or experimental errors. It represents a fundamental property of nature. The uncertainty in position Δx and the uncertainty in momentum Δp are related by: This means that making momentum as accurate as possible makes position inaccurate and vice-versa. If one is made zero the other has to be infinite.

The Heisenberg Uncertainty Principle Imagine that we create an electron beam and try to make it move in a horizontal straight line by inserting a metal with a small opening of size a. If the opening is very small, this means that we know very accurately the vertical position of the electron and the uncertainty in its vertical position will be no bigger than a, so Δx  a. BUT if the opening has the same size of the de Bröglie wavelength of the electron, the electron will diffract like waves of wavelength λ diffract when passing an aperture of size similar to λ. There is an uncertainty in the electron’s momentum Δp. The electrons will spread out with an angular size of 2θ.

The Heisenberg Uncertainty Principle The angle by which the electron is diffracted is given by: If we take the opening a as the uncertainty in the electron’s position in the vertical direction, we have Δp Δx  p = h This is a very simple explanation of where the uncertainty formula comes from. a electrons observed within this area θ ΔpΔp p But Therefore:

The Heisenberg Uncertainty Principle We can write Heisenberg's principle in terms of Energy and time as it also applies to these measurements. If a state is measured to have energy E with uncertainty ΔE, there must be an uncertainty Δt in the time during which the measurement is made, such that:

The Heisenberg Uncertainty Principle Heisenberg is pulled over by a policeman whilst driving down a motorway. The policeman gets out of his car, walks towards Heisenberg's window and motions with his hand for Heisenberg to wind the window down, which he does. The policeman then says ‘Do you know what speed you were driving at sir?', Heisenberg responds ‘No, but I knew exactly where I was.