Photons and Electrons, Part 2

Transcription

1 Photons and Electrons, Part 2 Erwin Schrödinger Albert Einstein

2 The Photoelectric Effect (1905) Wavelength dependence must be explained by the existence of quantized light. Albert Einstein Nobel Prize (his only one) awarded in 1921.

3 different metals have different thresholds because they possess different work functions, but the slope is the same...

4 Problem: The work function of sodium metal is ev, where 1 ev = x J. Find the wavelength of light capable of ejecting electrons from sodium metal with a speed of x 10 5 m s -1. Key equations: hν = hc/λ = Ekinetic + ϕmetal Step 1: Calculate Ekinetic in ev.

5 Problem: The work function of sodium metal is ev, where 1 ev = x J. Find the wavelength of light capable of ejecting electrons from sodium metal with a speed of x 10 5 m s -1. Key equations: hν = hc/λ = Ekinetic + ϕmetal Step 1: Calculate Ekinetic in ev. Ekinetic = ½mv 2 = (0.5)(9.109 x kg)(4.000 x 10 5 m s -1 ) 2 Ekinetic = x kg m 2 s -1 = x Joules Ekinetic = (7.288 x J)/(1.602 x J ev -1 ) Ekinetic = ev

6 Problem: The work function of sodium metal is ev, where 1 ev = x J. Find the wavelength of light capable of ejecting electrons from sodium metal with a speed of x 10 5 m s -1. Key equations: hν = hc/λ = Ekinetic + ϕmetal Step 2: Calculate hν in ev. hν = = ev note that we only keep 4 sig figs! We DO NOT report ev.

7 Problem: The work function of sodium metal is ev, where 1 ev = x J. Find the wavelength of light capable of ejecting electrons from sodium metal with a speed of x 10 5 m s -1. Key equations: hν = hc/λ = Ekinetic + ϕmetal Step 3: Calculate λ in nm. hν = ev = 1240./λ λ = 1240./2.735 = nm

8 Problem: The work function of sodium metal is ev, where 1 ev = x J. Find the wavelength of light capable of ejecting electrons from sodium metal with a speed of x 10 5 m s -1. Key equations: hν = hc/λ = Ekinetic + ϕmetal Step 3: Calculate λ in nm. hν = ev = 1240./λ λ = 1240./2.735 = nm A note on significant figures: 4 sig figs: v = x 10 5 m s -1 ; λ = nm 3 sig figs: v = 4.00 x 10 5 m s -1 ; λ = 453 nm 2 sig figs: v = 4.0 x 10 5 m s -1 ; λ = 450 nm

9 Quantum Mechanics: The Schrödinger Equation Erwin Schrödinger

10 An example: Particle in a Box (PIAB) H = p 2 /2m + V(x) Erwin Schrödinger V(x) for the PIAB.

11 Energy eigenstates: PIAB Energies and Wavefunctions Energy eigenstate wavefunctions:

12 classical mechanics says the following: 1. This particle can have any energy.

13 classical mechanics says the following: 1. This particle can have any energy. 2. It can be located anywhere, with equal probability.

14 classical mechanics says the following: 1. This particle can have any energy It can be located anywhere, with equal probability. 3. At any instant in time, this particle will have a particular position that can be specified with arbitrarily high precision.

15 quantum mechanics makes a completely different set of predictions: 1. This particle can only have certain allowed values of energy. For PIAB, E=0 is NOT allowed!

16 quantum mechanics makes a completely different set of predictions: 1. The particle can only have certain allowed values of energy. For PIAB, E=0 is NOT allowed! 2. We cannot know the exact trajectory and location of the particle in a given energy eigenstate. Only probabilities.

17 The wavefunctions for PIAB. Note that Ψ(0)=0 and Ψ(L)=0. This is a property of standing waves in general (e.g., guitar or violin strings,). Ψ(x)=0 for x <0 and x>l

18 What IS ψ(x)? The one dimensional eigenstate wavefunction has no direct physical significance itself. However, ψ 2 (x) is the probability of finding our particle in the interval from x to x + dx. "Born interpretation" (From Max Born, 1926). ψ 2 (x) is called the "probability density." example: the probability of finding the particle in the region, say, from 0 to L (inside the box) is: for all n = 1,2,3...

19 ψ 2 (x) is the probability density......here is what ψ 2 (x) looks like for n=2 (left) and n=1 (right)... e.g., for n=1, the particle prefers the center of the box!

20 classical mechanics says the following: 1. This particle can have any energy It can be located anywhere, with equal probability. 3. At any instant in time, this particle will have a particular position that can be specified with arbitrarily high precision.

21 quantum mechanics makes a completely different set of predictions: 1. The particle can only have certain allowed values of energy. 2. We cannot know the exact trajectory and location of the particle in a given energy eigenstate. Only probabilities. 3. There is an inherent uncertainty in the position and momentum of the particle: Heisenberg uncertainty principle

22 Heisenberg uncertainty principle There are many Heisenberg pairs of observables. The two most important are:

23 λ = h p p is exactly defined, but x is completely indeterminate. Δp=0; Δx?

24 λ = h p the summation of five sine waves, varying in momentum by 2% Δx Δp=2% wavepacket for a free particle.

25 Some easier nomenclature: States are sometimes just labelled with their quantum numbers

26 Absorption and Emission Spectroscopy Light will be absorbed or emitted only if E = hν The spectrum is QUANTIZED! E51=24G E41=15G E31=8G E21=3G G= h 2 /8mL 2

27 Atomic Emission Spectroscopy the atomic emission spectrum is the result of many atoms emitting light simultaneously. a spectrometer

28 Atomic Emission Spectroscopy The emission spectra of H, He and Hg are all quantized!

29 Atomic Emission Spectroscopy Therefore, AES tells us that the electron energy levels in atoms are quantized. Next up: the quantum mechanical calculation of the electron energy levels in a Hydrogen atom. Grotrian diagram for Hydrogen

30 The Photoelectric Effect (1905) Wavelength dependence must be explained by the existence of quantized light. Albert Einstein Nobel Prize (his only one) awarded in 1921.

31 The Photoelectric Effect and Photocurrents The photoelectric effect converts light into electrons. These electrons can be measured as a photocurrent. Using the power supply to stop the photocurrent is a way of measuring the maximum electron K.E.

32 The Photoelectric Effect and Photocurrents Photomultiplier tubes (PMTs) amplify the photocurrent from a alkali metal photocathode, creating pulses of a million electrons that can be detected.

33 The Photoelectric Effect and Photocurrents Photomultiplier tubes (PMTs) amplify the photocurrent from a alkali metal photocathode, creating pulses of a million electrons that can be detected.

34 The Photoelectric Effect and Photocurrents The photoelectron pulses from the PMT can be counted, yielding a measurement of photons per second. This detection method is called "photon counting".

35 The Photoelectric Effect and Photocurrents If the number of photoelectron pulses from the PMT becomes too large to detect individually, an average photocurrent can be measured. Photon flux can be calculated from the Radiant Sensitivity (A/W) of the PMT.