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
Wavefunctions of adjacent atoms 1 & 2 combine, so two possibilities: ψ1+ ψ2 or ψ1- ψ2
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?
Diatomic Molecule = interference of electron ‘waves’ (i.e. adding/subtracting)
OR
Bonds between atoms (cont.)
Bonding
Antibonding
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
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
Bonding
AntibondingBonds between atoms (cont.)
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
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 !
Charge with angular momentum – a magnetic dipole
Neutral, & suppose = ± µB
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
2. To every classical observable a there corresponds an operator
via Cartesian position x and momentum e.g. K.E. Energy
3. The only possible result of a measurement of an observable is an eigenvalue of the operator of that observable. Eigenvalue equation:
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
Key concept….
φ*φ dq = 1∫
A
−i ∂
∂x
p2
2m= −
2
2m∂2
∂x2 H = −
2
2m∂2
∂x2 +U x( )
Aφi = aiφi
ψ = cjφ j
j∑
4. If Φ is known, then the expectation value (value obtained on average) of observable a from operator  is given by
e.g. The 'probabilistic interpretation' of the state function
Suppose Φ is not eigenfunction of  but that Ψ is. i.e.
Then since we can write it follows that
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 and if then Ψ is not an eigenfunction of B nor is Φ an eigenfunction of A.
If then Ψ is an eigenfunction of B and Φ is of A. A-B Uncertainty requires
a = φ*∫ Aφ dq
x = ψ *∫ xψ dx = x ψ *ψ∫ dx
Aψ i = aiψ i
φi = ciψ i
i∑
a = φ* Aφ dq = ci
*∫i∑∫ ψ i
* Aciψ i dq = cii∑ 2
ai
Aψ = aψ Bφ = bφ AB − BA = A, B⎡⎣ ⎤⎦ ≠ 0
A, B⎡⎣ ⎤⎦ = 0
A, B⎡⎣ ⎤⎦ ≠ 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
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)
Q-bits: Quantum states: spins, Josephson etc., created in molecules, Si dopants, ion traps etc., addressed optically and electrically…. Technically challenging!