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School of Mathematical and Physical Sciences PHYS1220
PHYS1220 – Quantum Mechanics
Lecture 6
August 29, 2002
Dr J. Quinton
Office: PG 9
ph 49-21-7025
[email protected]
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School of Mathematical and Physical Sciences PHYS1220
Simple Harmonic Oscillator
Recall from mechanics
Classically,
U ( x) 
1 2 1
kx  m 2 x 2
2
2
The particle oscillates between x=A, where A – amplitude
1
1
The total energy of the system is Etot  KE  U  kA2  m 2 A2
2
2
Any value of A (and hence Etot) is allowed
Total Energy is zero if the particle is at rest at x=0
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
Quantum Mechanics
T.I.S.E.
U(x)
d 2 ( x) 1
 m 2 x 2  ( x)  E ( x)
2
2m dx
2
2
d 2 ( x)  2mE   m  
  2   
  ( x )
dx 2
 
 

2
Analysis complicated, so guess
 ( x)  BeCx Gaussian function
x
2
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School of Mathematical and Physical Sciences PHYS1220
Simple Harmonic Oscillator II
Substituting into T.I.S.E.
the guess corresponds with
the ground state
 1 ( x)  Be
C
m
2
E
1

2
classical
 m  2

x
2 
This is only one solution. The
excited states are polynomials
multiplied by an exponential
(Gaussian) function
 n ( x )  f ( x )e
Quantum
 Cx 2
Penetration into barrier
occurs
Plot of probability shows large
variation from classical
predictions
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Simple Harmonic Oscillator III
Energy is quantised
1

En   n    , n  0,1, 2,3,...
2

U(x)
n=0 is ground state, with zero-point
energy
1
E0 
2

E4  9  / 2
Result justifies Planck’s hypothesis
regarding vibrational energies
E3  7  / 2
E  nhf , n  1, 2,3,...
Correspondence principle applies
because lim En  En 1  
E2  5  / 2
E1  3  / 2
E  
E0   / 2
n 
x
but is negligible compared to the
actual energy
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Barrier Potentials
According to QM, wave functions can penetrate the walls of the
potential (provided that U is finite) and there is a non-zero
chance that the particle exists inside the wall region.
Question: What if the wall is not infinitely thick?
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Tunnelling
If we have a sufficiently energetic
electron, and a thin wall or barrier, the
electron may actually tunnel through
the barrier.
The solution to the bound particle in a
finite well had the wavefunction
decaying exponentially in the wall. If
the wall is thin, there is a non-zero
amplitude to the wavefunction at x=L.
2
After the wall   0
Reflection and Transmission
coefficients may be developed such
that R+T=1.
If T<<1 then: T  e  2GL
G
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2m(U 0  E )
2
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School of Mathematical and Physical Sciences PHYS1220
Tunneling - II
According to QM, the particle exists on
both sides of the barrier.

It just has a different probability of
being on one side than the other
It is not until you go to measure the
particle (and collapse its wave
function) that you actually know
which side of the barrier it is on
In going through the wall, no energy
is lost (the particle’s energy is still E).

Remember the amplitude is related to
the probability, not the energy. The
energy is related to the frequency
(wavelength)
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Example
Question: A 50 eV electron approaches a square barrier
potential 70 eV high and (a) 1.0 nm thick (b) 0.10 nm thick.
What is the probability that the electron will tunnel through?
Answer:
(a) First convert to SI units
U 0  E  (70eV  50eV) 1.6x10 19 J/eV  3.2x10-18 J
2(9.11x1031 kg x 3.2x1018 J
2GL  2
(1.0x109 m)  46
34
1.06x10 J.s
T  e 2GL  e 46  1.1x1020
which is extremely small
(b) for L=0.1nm, 2GL = 4.6
T  e2GL  e4.6  0.010
so by decreasing the barrier width by a factor of 10, the
probability of tunnelling has increased by 18 orders of magnitude
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Applications of Quantum Mechanics
~ 30% of the US national Gross Domestic Product (GDP) is
directly due to applications of Quantum Mechanics


Semiconductor industry
Digital communications with light/optic fibres
Quantum mechanics has applications in many diverse areas,
ranging from (but not limited to)
•
•
•
•
•
Inert gas signs
Semiconductors
Lasers
Microwave ovens
MRI imaging
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•
•
•
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Transistors
Quantum dots
Conducting polymers
Quantum computing
Scanning Tunnelling Microscope
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Inert Gas Signs
Work on discharge lamp principle
Gas is inert (or mixture) Ne, Ar, Kr
Emission line spectrum has discrete
wavelengths, usually one colour
dominates (due to higher transition
probability)
By varying the pressure, it is possible
to alter which electron states
dominate, and hence alter the
emission spectrum and therefore tune
the sign’s colour to some degree
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Scanning tunnelling microscopy



Vary the position of the tip above the surface to keep a constant
tunnelling current and then plot the position control voltage against
sample x,y dimension.
Binning and Rohrer shared the 1986 Nobel prize.
If this technology were used instead of optical techniques, a cd could
store >1012 bytes of information.
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STM image of Au on mica at T~70K
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Scanning Tunnelling Microscopy
Reproducible Measurement of Single-Molecule
Conductivity
X. D. Cui, A. Primak, X. Zarate, J. Tomfohr, O. F. Sankey, A. L.
Moore, T. A. Moore, D. Gust, G. Harris, S. M. Lindsay SCIENCE
VOL 294 19 OCTOBER 2001
Writing with Atoms
(achieved by pushing them
around with the tip)
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School of Mathematical and Physical Sciences PHYS1220
Quantum Computing
Atomic scales are very small, device miniaturisation not an issue
compared with current technology
Quantum bits (qubits) are atomic or molecular spin states

‘up’ or ‘down’
 and 
Bit states are controlled by light
Registers are made from several finite well devices
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Final Thought
"Anyone who isn't shocked by
quantum theory has not understood it."
Neils Bohr, ~75 years ago
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