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!