1. 7.5 Quantum Theory & the Uncertainty Principle “But Science, so who knows?!”

2. Objectives Describe emission and absorption spectra and understand their significance for atomic structure; Explain the origin of atomic energy levels in terms of the ‘electron in a box’ model; Describe the hydrogen atom according to Schrӧdinger; Do calculations involving wavelengths of spectral lines and energy level differences; Outline the Heisenberg uncertainty principle in terms of position-momentum and time-energy.

3. Atomic spectra Hydrogen gas heated to high temps/exposed to high electric field ⇒ glows (EMITS light) Analyze light by sending it through a spectrometer (splits light into component wavelengths) λ = 656 nm (H α ) red λ = 486 nm (H β ) blue-green

4. Emission spectrum – the spectrum of light that has been emitted by a gas Bright Lines

5. Absorption Spectrum – the spectrum of light that has been transmitted through a gas. White light (all wavelengths) passed through hydrogen gas, then analyzed with a spectrometer. The dark lines in the absorption spectrum are at the exact same wavelengths as the colored bright lines in the emission spectrum.

6. Atomic Spectra

8. The ‘electron in a box’ model Since the electron is confined to the box, it is reasonable to assume that the electron wave is zero at both edges of the box.

9. The ‘electron in a box’ model In addition, since the electron cannot lose energy, it is also reasonable to assume that the wave associated with the electron in this case is a standing wave.

10. The ‘electron in a box’ model This result shows that, because we treated the electron as a standing wave in a ‘box’, we deduce that the electron’s energy is ‘quantized’ or discrete, i.e. it cannot have any arbitrary value.

11. The ‘electron in a box’ model This model gives us a discrete set of energies Not a realistic model for an electron in an atom, but it does show the discrete nature of the electron energy when the electron is treated as a wave; points toward the correct answer.

12. The Schrӧdinger Theory 1926 – Austrian Physicist Erwin Schrӧdinger Assumes as a basic principle that there is a wave associated to the electron (like de Broglie), called the wavefunction, ψ(x,t). The wavefunction is a function of position x and time t. Given the force that act on an electron, it is possible, in principle, to solve a complicated differential equation obeyed by the wavefunction (the Schrӧdinger equation) and obtain ψ(x,t).

13. The Schrӧdinger Theory For example, there is one wavefunction for a free electron, another for an electron in the hydrogen atom, etc. The interpretation of what ψ(x,t) really means came from German physicist Max Born. He suggested that |ψ(,)| (the square of the absolute value of ψ(x,t) can be used to find the probability that an electron will be found near position x at time t.

14. The Schrӧdinger Theory The theory only gives probabilities for finding an electron somewhere – it does not pinpoint and electron at a particular point in space; a radical change from ordinary (classical) physics where objects have well defined positions. When the Schrӧdinger theory is applied to the electron in a hydrogen atom it gives results similar to the simple electron in a box example of the previous section.

15. The Schrӧdinger Theory

17. Example

18. Calculate the wavelength of the photon emitted in the transition from n=3 to n=2.

19. The Schrӧdinger Theory The variation of the probability distribultion function (pdf) with distance r from the nucleus for the n=1 (lowest) energy level of the hydrogen atom. The height of the graph is proportional to |ψ(,)|. The shaded area is the probability for finding the electron at a distance from the nucleus between r=a and r=b.

20. Assignment 3,4,5,7

21. The Heisenberg Uncertainty Principle Discovered 1927 - Named after Werner Heisenberg (1901 – 1976); one of the founders of quantum mechanics Founding idea: wave-particle duality – particles sometimes behave like waves and waves sometimes behave like particles, so that we cannot cleanly divide physical objects as either particles or waves

22. The Heisenberg Uncertainty Principle

23. Imagine: electrons emitted from a hot wire in a cathode ray tube (crt) and we try to make them move in a horizontal straight line by inserting a metal with a small opening of size a. we can make the electron beam as thin as possible by making the opening as small as possible – electrons must be somewhere within the opening so Δx < a. a should not be on the same order of magnitude as the de Broglie wavelength of the electrons to avoid diffraction.

24. The Heisenberg Uncertainty Principle Here, too, the electron will diffract through the opening ⇒ some electrons emerge in a direction that is no longer horizontal. We can describe this phenomenon by saying that there is an uncertainty in the electron’s momentum in the vertical direction of magnitude Δp

25. The Heisenberg Uncertainty Principle

29. Assignment 8,11,15