Quantum Physics II

UNCERTAINTY PRINCIPLES

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

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

Pg 5 – 7 WHAT IS WAVE FUNCTION?

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 

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

What is wave function? Refer to Dr. Quantum movieclip http://www.youtube.com/watch?v=DfPeprQ7oGc

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

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

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!

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

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

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

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

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 |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)

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

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

Pg 8 – 9 QUANTUM TUNNELLING

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

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

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

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

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

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

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

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

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

APPLICATION OF TUNNELLING Pg 10 – 12 APPLICATION OF TUNNELLING

Scanning tunnelling microscope

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

Alpha decay

A-LEVEL QUESTIONS

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

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

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.

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.

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

EXTRA QUESTIONS

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]

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

H3 The width of a spectral line of wavelength 400 nm is measured to be 10-14 m. What is the average time the atomic system remains in the corresponding energy state? [4.24 x 10-9 s]

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 10-14 m [0.013 MeV]

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 10-16 s, 5.33 x 10-13 m]