1. MS310 Quantum Physical Chemistry
Ch 6. Commuting and Noncommuting Operators and the Surprising Consequence of Entanglement
Applied simple quantum mechanical framework in
real experiment. (Stern-Gerlach experiment)
- Noncommuting operators concerning position and
monentum. ( Heisenberg uncertainty principle)
Particle in a 3-D box and Quantum computers
MS310 Quantum Physical Chemistry

2. MS310 Quantum Physical Chemistry
6.1 Commutation relations
There are 2 observables a and b, corresponding operator
We can think two cases.
1) Measurement A first, B after
2) Measurement B first, A after
If ψn(x) is eigenfunction of operator A(no state change)
Also, if ψn(x) is eigenfunction of operator B(no state change also)
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If ψn(x) is eigenfunction of operator B
Also, if ψn(x) is eigenfunction of operator B
ψn(x) is eigenfunction of operator A and B both
→ result is independent of the order of measurement
Otherwise, two results are different.
Two operator A and B have a common set of eigenfunction
→ must satisfy the commutation relation
(f(x) is arbitrary function) and if it satisfied, A and B commute.
Notation :
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ψn(x) is eigenfunction of operator A : no change after the measurement the observable a
If satisfied : state ψn(x) is not change by the two measurement the observable a and b
→ ‘can measure simultaneously and exactly two observable a
and b’
Ex) 6.1
Momentum and a) kinetic energy b) the total energy can be known simultaneously?
Sol)
Use the commutator a)
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Momentum and kinetic energy is commute.
Therefore, momentum and kinetic energy can be known simultaneously. b)
Therefore, we cannot be known the momentum and total energy simultaneously.
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6.2 The Stern-Gerlach experiment
Consider the dipole in the inhomogeneous magnetic field.
In this situation, dipole orient and deflect to the magnetic field.(parallel and antiparallel to the magnetic field)
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Stern and Gerlach did the experiment.
condition : external magnetic field applied to the Ag beam
Result : Ag beam split two beams.
→ 2 eigenvalues of measure the z-component of the magnetic momentum
We write operator of measurement the z-component of the magnetic momentum as A, wavefunction of one spin as α, other spin as β.
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Cannot specify the value of c1 and c2.
However, ratio of two beam is 1 by the individual measurement.
Measure the direction of x-component of magnetic momentum of the beam of state α : ‘split’ 2 beams!
(in this case, write the operator : B and wavefunction : γ, δ)
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Then, operator A and B commute? No.
If two operator commute → eigenfunctions of 2 operators same
→ result of second measurement is only 1 state. Why?
‘after’ the first measurement, wave function collapse to only 1 measured state. Second measurement measures the ‘collapsed’ state, one of the eigenfunctions of the first measurement. If two operator commute, second measurement measures the eigenfunction of operator B, and result must be one state.
However, result of second measurement also split to 2 beams.
Therefore, measurement of z-component of magnetic moment and measurement of x-component of magnetic moment do not commute.
Result : Ag atom doesn’t have well-defined values for both μz and μx simultaneously.
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6.2.1 The history of the Stern-Gerlach experiment
Experiment did in 1921
Ag beam generation : oven in a vacuum chamber was collimated by 2 narrow slits of 0.03mm width
Beam passed into inhomogeneous magnetic field 3.5cm and impinged on a glass plate.
1 hr operation in this experiment.
How can see the Ag?
→ ‘sulfur’
Sulfur reacts to Ag and makes Ag2S.
Ag2S : black, and it can see less than 10-7 mol of Ag
→ reason of successful experiment
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6.3 The Heisenberg uncertainty principle
Heisenberg uncertainty principle : ‘cannot know simultaneously position and momentum of particle’
It starts that position and momentum do not commute.
Wavefunction of free particle : Ψ(x,t)=Aexp[i(kx – ωt – φ)]
Set φ=0 and t=0 : focus on spatial variation of ψ(x)
We normalized wavefunction into finite interval [-L,L]
Probability of x=x0 : P(x0)dx=ψ*(x0)ψ(x0)dx
L → ∞ : probability approaches to 0! → no data of position
It gives this result : if we know momentum exactly, position is completely unknown
Similarly, if we know position exactly, momentum is completely unknown
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Consider the superposition of plane waves of very similar wave vectors
See the case of m=10(21 waves superposition)
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Wave vector k0 : 7.00 x m
Case of 21 waves, peak of the probability : 0 , x m
→ range of probability exist decrease(wave packet) : probability localized into finite interval → uncertainty of position increase.
Superposition of a lot of plane wave : cannot know exactly the wave vector of particle → ‘uncertainty’ of momentum
More wave superposition occurs, uncertainty of particle decrease, but uncertainty of momentum increase!
Consider the ∆k << k0, momentum of wavefunction is given by
It means, range of momentum increase when m increase.
Finally, we can obtain Heisenberg uncertainty principle
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Text p.88
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Text p.89
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Text p.89
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6.4 The Heisenberg uncertainty principle expressed
in terms of standard deviation
Heisenberg uncertainty principle can be written in the form
σx,σp : standard deviation of position and momentum
This 4 values are defined by postulate 4.
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Consider the particle in a box
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n=1 : minimum → uncertainty principle satisfied for all n
Relative uncertainty in x and p
when n→∞, uncertainty of position increases.
→ related to probability of finding particle is equal everywhere case of large n.
However, uncertainty of momentum is independent to n.
→ uncertainty of momentum can be negligible.
But, it is not enough : there are 2 p values when p2 determined.
Solution : change the wavefunction as eigenfunctions of momentum operator
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relative probability density of wave vector
Momentum approaches to classical value when n increase! → relative uncertainty of momentum ‘decrease’ as n increase!
MS310 Quantum Physical Chemistry

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6.5 A thought experiment using a particle in
a 3-dimensional box
We do thought experiment by these steps.
1) A particle in a box
→ know the wavefunction of particle
2) Insert barrier
→ tunneling probability decrease in middle region
3) Move apart : separate to 2 boxes
→ wavefunction represent by
(each function satisfies the particle in a box and can assume a=b) : superposition state
4) Look in box(measurement one of the boxes)
→ we can see only particle is in the left or right : ‘rapidly decay of superposition state when measurement occurs’
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6.6 Entangled states, teleportation, and
quantum computer
Consider the case in 6.5, the particle(single particle) is in the superposition state.
Also, this wavefunction is not an eigenfunction of position.
If two quantum particles are strongly coupled : entangled state
Beam of photons is incident on transparent cystalline BaTiO3.
→ Only 2 direction of electric field vector of photon : Horizontal(H) and Vertical(V) : polarization state
Probability of horizontal and vertical is same by the measurement, and ‘if polarization of first photon measured, the other polarization will be measured exactly!’
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Wavefunction can be described by
This wavefunction is not an eigenfunction of single particle operator, and measure the single particle have no meaning because this system is ‘entangled’ state.
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How can use this result : ‘teleportation’
There are a pair of entangled photons, Alice has photon A and Bob has photon B.
Consider the photon A is entangled to photon X. It means photon A and photon X is orthogonal.
Photon B is entangled to photon A and photon B must be orthogonal to photon A. Therefore, state of photon B is same as photon X, the message of Alice : teleportation!
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More interesting application : quantum computer
Classical computer(our PC) : bit
n bit memory : 000…0 to 111…1 : 2n state
Quantum computer : qubit
Use the superposition of different quantum state.
In photon system, H and V can be correspond to 0 and 1
Superposition state → qubit
n-qubit system : entangled 2n state
In a bit, 2M state stored in length M. However, in a qubit, 2M state stored in M-qubit, only one superposition state!
Therefore, 2M simultaneous calculation can be parallel in M-qubit quantum computer.
If M=30, 1030 calculation can be parallel, and it expects the speed of calculation improve surprisingly.
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Summary
The Heisenberg uncertainty principle limits the degree to which observables of noncommuting operators can be known simultaneously.
The Stern-Gerlach experiment clearly demonstrates that the prediction of quantum mechanics is obeyed at the atomic level.
Entanglement is the basis of both teleportation and quantum computing
MS310 Quantum Physical Chemistry