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Quantum Physics II
UNCERTAINTY
PRINCIPLES
What are uncertainty principles?

In QM, the product of uncertainties in
variables is non-zero
◦ Position-momentum
◦ Energy-time E t  h
4
h
px x 
4
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=Df
PeprQ7oGc
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

Intensity, I  kx02


Using principle of superposition

Net displacement,xR  x1  x2
The intensity at any constructive
interference region is

IR 
kxR2

 k x01  x02

2
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

Iwav e  x01  x02
Probability  1  2


2
2
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

Iwav e  x02
2
Prob. density function     
Probability of finding particle between x1 and x2

is
x2
2
Px1  x  x2    dx

x1

What is wave function?

Wave treatment of particle is basically
statistical
◦ Conventional statistics
x 




xf x dx
x2 
◦ Quantum mechanically


x 



  xdx
x2 





x 2f x dx


  x 2dx
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
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 ||2

In quantum
mechanics 2 is
proportional to the
probability of finding
the particle at a
given location.
||2
Pg 8 – 9
QUANTUM
TUNNELLING
Potential barrier

Consider the GPE of a mass m near
Earth’s surface
GPE
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 > 0
Ek = ET - EP
EP = ET
Ek = 0
h
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
EK
ET
EP = ET
Ek = 0
Turn back!
EP = 0
Ek = ET
h0
h
Potential barrier

The potential confines the particle within
a region, it is not allowed beyond h0
PE
EK
ET
h0
h
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

R T  1


T  exp 2kL
where k 

2m U  E
h

Wave function in potential barrier

Some examples of wave functions in wellknown potential
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.

energy
level of
electron
in atom
Distance from
centre of 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
PE of
electron
Classically, an electron could
Energy of
never exist on the right of
electron
the potential barrier because
x
x
it does not have sufficient
energy
 If the electron is treated as a wave and applying
Schrodinger equation, its wave function

1
2
◦ 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
x
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
h
E t 
4
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 onedimensional 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]