1. Quantum Physics II

2. UNCERTAINTY PRINCIPLES

3. What are uncertainty principles?
In QM, the product of uncertainties in variables is non-zero
Position-momentum
Energy-time
Intrinsic imprecision, not due to measurement limits
Measurements (e.g. px and x) on identical systems do not yield consistent results

4. Is vacuum really empty?
Energy can be “borrowed” from nothing on the condition that this energy is returned within a certain time governed by the energy-time uncertainty principle
This borrowed energy becomes the mass of particles (E = mc2)
The larger the energy “borrowed”, the shorter its lifetime
The larger the mass of the particle created, the shorter is its lifetime

5. Pg 5 – 7
WHAT IS WAVE FUNCTION?

6. Definition*
Wave function of a particle is a complex quantity that is the probability amplitude of which the absolute square gives the probability density function for locating the particle within regions of space
The rest of this section is not in your syllabus! So don’t panic if you don’t understand at all 

7. What is wave function? Recap: Wave experiments Double- slit experiment
Single slit experiment

8. What is wave function?
Refer to Dr. Quantum movieclip

9. What is wave function? If e- are used instead of light
Pattern builds up spot by spot
Particle-like
Distribution is interference pattern!
Wave-like
Intensity represents probability of electron landing on that region

10. What is wave function? Let’s return to interference of waves
The intensity at a region is
Using principle of superposition
The intensity at any constructive interference region is

11. What is wave function?
Since e- beams thru 2 slits interfere and the intensity at a region represents probability
Rewriting the interference relation for e-
What passes thru each slit must be the probability amplitude!

12. What is wave function?
Wave function is a complex (not a pun) quantity interpreted as the probability amplitude
The absolute square gives the probability density function
Probability of finding particle between x1 and x2 is

13. What is wave function?
Wave treatment of particle is basically statistical
Conventional statistics
Quantum mechanically
Do you notice the similarity?
Hence the absolute square of wave function gives the probability density function

14. Electrons through double slits
Assume that the particle (eg electron) can be represented by a mathematical expression eg a wave function  (which could be complex) and also assume that the intensity profile of the interference pattern (eg the numbers of electrons detected per second) can be expressed by the square of the absolute value of this wave function |  |2

15. If slit 1 is opened (slit 2 closed), then we can represent the wave function of the electrons passing through slit 1 as 1 and therefore the intensity profile is | 1 |2
If slit 2 is opened (slit 1 closed), then we can represent the wave function of the electrons passing through slit 1 as 2 and therefore the intensity profile is | 2 |2
If we open slit 1 for half the time (slit 2 closed) and then slit 2 for half the time (slit 1 closed) then the intensity profile will be | 1 |2 + | 2 |2

16. The intensity profile is then | 1 + 2 | 2 =
If we open both slits then the electron wave functions are superimposed (similar to light). The combined wave function is then 1 + 2
The intensity profile is then | 1 + 2 | 2 =
| 1 |2 + | 2 | (1 . 2)
This is different from the previous case of opening one slit 50% of the time, ie | 1 |2 + | 2 |2
The term 2 ( 1 . 2 ) represents the interference term.
Note that if the wave functions are complex, then | 1 |2 =  1  1* (where  1* is the complex conjugate)

17. Where have you seen wave functions?
Electron clouds and orbitals in Chemistry
Orbitals are square of wave functions!

18. The probability density |Y|2
||2
In quantum mechanics 2 is proportional to the probability of finding the particle at a given location.

19. Pg 8 – 9
QUANTUM TUNNELLING

20. Potential barrier Consider the GPE of a mass m near Earth’s surface
GPE = mgh h

21. Potential barrier
If a particle have total energy ET is projected upwards from the ground (GPE define as 0)
GPE
Turn back at this height
GPE = mgh ET
EP = 0
Ek = ET EK
EP = ET
Ek = 0 h
EP > 0
Ek = ET - EP

22. Potential barrier
Consider an arbitrary PE for a mass m PE h

23. Potential barrier
If the mass m has total energy ET, is projected from h = 0 with EK PE ET EK
EP = ET
Ek = 0
Turn back!
EP = 0
Ek = ET h h0

24. Potential barrier
The potential confines the particle within a region, it is not allowed beyond h0 PE ET EK h h0

25. Potential barrier
Potential barrier are gravitational, electrical in nature
Related to potential energy
Invisible, not a physical obstacle!
It is a barrier when the potential energy of the particle at a particular position(s) in space is larger than the particle’s energy
ie, the particle cannot reach such position(s) given its current total energy

26. Quantum tunnelling
Classically particle cannot move into and past the region of the potential barrier because its energy is not sufficient

27. Quantum tunnelling
The wave treatment of particle allows a finite probability in/beyond the region of the potential barrier
reflected
transmitted

28. Wave function in potential barrier
Some examples of wave functions in well- known potential

29. APPLICATION OF TUNNELLING
Pg 10 – 12
APPLICATION OF TUNNELLING

30. Scanning tunnelling microscope

31. Scanning tunnelling microscope
Potential barrier is the gap
Tunnelling when the gap is small enough
Tunnelling current
Small applied p.d. for a fix current direction
Refer to Eg 9 for modes of operation

32. Alpha decay

33. A-LEVEL QUESTIONS

34. Q1 – SP07/III/8d
A electron in an atom may be considered to be a potential well, as illustrated by the sketch graph
Explain how, by considering the wave function of the electron, rather than by considering it as a particle, there is a possibility of the electron escaping from the potential well by a process called tunnelling.
Distance from centre of atom
energy level of electron in atom

35. Q1 – Solution
Barrier width
Classically, an electron could never exist outside the potential barrier imposed by the atom because it does not have sufficient energy
If the electron is treated as a wave and applying Schrodinger equation, its wave function
is sinusoidal with large amplitude between the barrier
decays exponentially within the barrier
is sinusoidal with a much smaller amplitude outside the atom
The square of the wave function gives a small but finite probability of finding the electron outside the atom

36. Q2 – SP07/III/8e
The process in Q1 is used in a scanning tunnelling microscope, where magnifications of up to 108 make it possible to see individual atoms. Outline how these atomic-scale images may be obtained.

37. Q3 - N07/III/7e
Show, with the aid of a diagram, what is meant by a potential barrier. Discuss how the wave nature of particles allows particles to tunnel through such a barrier.

38. Q3 – Solution
Energy of electron
PE of electron x1 x2 x
Classically, an electron could never exist on the right of the potential barrier because it does not have sufficient energy
If the electron is treated as a wave and applying Schrodinger equation, its wave function
is sinusoidal with large amplitude before the barrier
decays exponentially within the barrier
is sinusoidal with a much smaller amplitude after the barrier
The square of the wave function gives a small but finite probability of finding the electron to the right of the barrier

39. EXTRA QUESTIONS

40. H1
What is the uncertainty in the location of a photon of wavelength 300 nm if this wavelength is known to an accuracy of one part in a million?
[23.9 mm]

41. H2
If we assume that the energy of a particle moving in a straight line to be mv2/2, show that the energy-time uncertainty principle is given by

42. H3
The width of a spectral line of wavelength 400 nm is measured to be m. What is the average time the atomic system remains in the corresponding energy state?
[4.24 x 10-9 s]

43. H4
A particle of mass m is confined to a one- dimensional line of length L
Find the expression of the smallest energy that the body can have
What is the significance of this value?
Calculate the minimum KE, in eV, of a neutron in a nucleus of diameter m
[0.013 MeV]

44. H5*
If the energy width of an excited state of a system is 1.1 eV and its excitation energy is 1.6 keV,
what is the the average lifetime of that state?
what is the minimum uncertainty in the wavelength of the photon emitted when the system de-excites?
[2.99 x s, 5.33 x m]