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Quantum Physics Lecture 11
Bonding between atoms
Uncertainty principle revisited
Stern-Gerlach experiment
- Measurement
Formal postulates of Quantum Mechanics
Superposition of states - Quantum computation & communication
Bonds between atoms
Isolated atom in ground state Ψ
e.g. H atom 1s state
Probability of finding electron is ∝ ⎮ψ⎮2
Note: Wavefunctions can be +ψ or –ψ
What happens when two atoms
approach each other?
Wavefunctions of adjacent atoms 1 & 2 combine,
so two possibilities: ψ1+ ψ2 or ψ1- ψ2
Bonds between atoms (cont.)
OR
Diatomic Molecule = interference of electron ‘waves’
(i.e. adding/subtracting)
Antibonding
Bonding
Electron more likely to be
between nuclei compared to
isolated atom - saves
electrostatic energy
⇒ Bonding state
Electron is removed from region
between nuclei compared to
isolated atom
Costs energy.
Anti-Bonding state
Overall energy saving (= bonding) if electrons go into bonding state
e.g. OK for H2+ or H2 molecules.
Electrons are ‘shared’ – covalent bond
Bonds between atoms (cont.)
Antibonding
Bonding
Note for He2 (4 electrons), Pauli principle means two e’s in
antibonding state as well as bonding state
so no overall energy saving
(inert gases – no bond - no He2)
Mid-periodic table elements (half-filled orbitals) tend to have strongest
bonds (e.g. melting points. etc.)
ψ is ‘periodic’ inside atom & decaying outside – ‘barrier’ between atoms
but electrons move between atoms by tunnelling.
➞ Exponential variation of energy of interaction with separation
– Interatomic forces
Two-slit experiment
Observe:
- Close one slit (i.e. the particle must go through the other) ⇒lose the 2-slit diffraction pattern!
-  Single particle causes single point of scintillation
⇒ pattern results from addition of many particles!
- Pattern gives probability of any single particle location G.I.Taylor
Low intensity beam
Stern – Gerlach experiment
Neutral, & suppose = ± µB
Charge with angular momentum
– a magnetic dipole
Strong non-uniform magnetic field. Produces net force on dipole.
Direction of force depends on orientation of dipole and field gradient
Random orientation of dipoles fed in – So classically, expect a range of deflections.
Actually get two deflections ONLY!! – ‘up’ and ‘down’ states !
Sequential 90˚ Stern-Gerlach
↑ beam input
split into two:
← and →
Process of measuring dipole in z-direction direction forces spins into
one of the two possible states that can result from measurement!
For 90˚, input spin has equal probability of giving either output spin
Can think of as a superposition of the possible output states…
Triple S-G on z, y, z axes
3rd SG gives z-split again
With equal probability
From single y-axis spin
Uncertainty principle – cannot know
whether up or down will result
Note complete loss of information of first z split information after
passing through (orthogonal) y-split!
Actual state is changed by measurement…..
How to provide a formalism for these results?
And What is the state function before the measurement? Postulates of QM
1.  The state of a system is completely described by a state function Φ(q1, q2 ..... qn)
where the system has variables (coordinates) q1,.... qn
NB. Φ is not an observable, is single valued and can be normalised by ∫ φ *φ dq = 1
2.  To every classical observable a there corresponds an operator
via Cartesian position x and momentum
 ∂ e.g. K.E. p̂ 2
 2 ∂2
 2 ∂ 2 Energy
=−
Ĥ = −
+U x
−i
2m
2m ∂x 2
2m ∂x 2
∂x
()
3.  The only possible result of a measurement of an observable is an eigenvalue of the
operator of that observable.
Eigenvalue equation: Âφ = a φ
i
i i
In general there will be a complete set of functions Φi which satisfy the eigenvalue equation.
e.g. the set of sin(nkx) & cos(nkx) functions of the’ waves in a box’ - cf Fourier components
Any other function can be expressed as a linear combination of these functions ψ = ∑ c jφ j
j
Key concept…. 4.  If Φ is known, then the expectation value (value obtained on average) of
observable a from operator  is given by a = ∫ φ * Âφ dq
*
*
e.g. x = ∫ ψ x̂ ψ dx = ∫ x ψ ψ dx The 'probabilistic interpretation' of the state function
Suppose Φ is not eigenfunction of  but that Ψ is.
Then since we can write φi =
∑
i
i.e. Âψ i = aiψ i
ciψ i it follows that a = ∫ φ * Âφ dq = ∑ ∫ ci*ψ i* Âciψ i dq = ∑ ci ai
2
i
i
So |ci|2 is the probability of ai being the actual result measured, out of all those possible.
5. Immediately after measuring the result of Â, the system is in a state which is an
eigenfunction of Â.
If the system was not in an eigenstate of  before the measurement then the measurement changes the state of the system! NB. For Âψ = aψ
If
and
B̂φ = b φ
if ÂB̂ −
B̂Â = ⎡⎣ Â, B̂ ⎤⎦ ≠ 0 then Ψ is not an eigenfunction of B
nor is Φ an eigenfunction of A. ⎡ Â,
⎤
then Ψ is an eigenfunction of B and Φ is of A. ⎣ B̂ ⎦ = 0
A-B Uncertainty requires ⎡⎣ Â, B̂ ⎤⎦ ≠ 0
Provided an actual measurement of a variable is not made, a
system can be in a state which is a superposition of states which
would result from a measurement of that variable…
Uncertainty – which state will actually be the result when measured?
Recall particle diffraction. Many measurements vs. single measurement AND…. More exotic applications…
Quantum cryptography: Information sent by state e.g. single photon polarisation
Interception to measure the state changes it.
Eavesdropping can be detected!
In principle - unbreakable
In practice - resilient
implementation is difficult
Quantum computation: instead of binary 1 – 0 can have a ‘q-bit’
which is a superposition of states
Q-bits: Quantum states: spins, Josephson etc., created in molecules, Si dopants,
ion traps etc., addressed optically and electrically…. Technically challenging!
Computation using q-bits can allow many
combinations to be calculated simultaneously.
Vary rapid scaling for large calculations. Potential applications include factoring of large
numbers…..
Problem: keeping the q-bits stable against
unintended interactions (decoherence)