Matricies and Waves

7/24/2019 Matricies and Waves

1/13

1.2 Matrices, waves and a statistical

interpretation

How should we describe the quanitzation weveobserved?

How can we interpret the wave-like behaviour ofparticles such as electrons?

What are the implications?


2/13

What Heisenberg did first

Heisenberg wanted to develop adescription of reality based on

the discrete properties of sys-

tems: e.g. energy, angular mo-

mentum.

He realised that what we observe

is thetransition,E=E2E1,

rather than E1 or E2 directly.


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Heguessedthat the new quanitities should depend ontwostates,

and suggested that familiar quantities such as x and p should be

replaced as x x11, x12,...,x21, x22,...Luckily, Born and Jordan recognized this as a matrix:

x=

x11 x12 x13 ...

x21 x22 x23 ...

x31 x32 x33 ...... ... ... ...


4/13

They also found the rule,

pxxp=

h

i I

which we will come back to later.

In collaboration with Heisenberg, they showed that this condition

leads to the quantization ofE, L already observed through line

spectra, the Franck-Hertz and Stern-Gerlach experiments.

I was discouraged, if not repelled, by what appeared

to me a rather diffiult method of transcendental

algebra, defying any visualization.

Schrodinger


5/13

So then what are matter waves???

How should we interpret the wave-like behaviour?This is a deep question, and historically has been the cause of a

great deal of debate. The winner (at the moment, at least) is

essentially Max Born, who proposed the statistical interpre-

tation of matter waves.

Assumption: The state of a particle is represented

by a complex function(r, t) such that ||2dV is

the probability of finding that particle at the time

t in the volume element

dVat the point

r.


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The Born interpretation

Born, talking about his quantum theory of scattering, said:

One does not get an answer to the

question, What is the state after

collision? but only to the question,

How probable is a given effect of

the collision? From the standpoint

of our quantum mechanics, there is

no quantity which causally fixes the

effect of a collision in an individualevent.


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The statistical interpretation

Some of the features of a statistical interpretation:normalization: we usually try to choose (r, t) so that

||2dV =

dV = 1

we can calculate the mean positionas

x=

x||2dV

in fact, the mean of any function ofx, f(x), is

f(x)= f(x)||

2

dV

and we can define a variance, 2

2 =(x)2= x2 x2


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Wavefunctions

The complex function (x, t) which describes the particles prob-ability distribution is awavefunction.

This wavefunction must be gov-

erned by a wave equation analo-gous to the classical wave equa-

tion Schrodinger formulated

such an equation in 1927.

The more I ponder about the

physical part of Schrodingers

theory, the more disgusting it

appears to me. Heisenberg


9/13

Support for a probability interpretation

One of the unique features of quantum physics, for which there is noclassical equivalent, is the phenomenon oftunnelling. Examples

include: transport between coupled quantum dots, nuclear fusion,

electron microscopes, tunnelling diodes,decay ...

The wavefunction of the parti-

cle is extended: some of the am-

plitude (i.e. theprobability den-

sity) is outside the Coulomb bar-

rier and sothere is a chance thatthe particle will be outside the

nucleus.


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Einstein, de Broglie, Schrodinger, Planck, Bohm all vehemently

disagreed with the Copenhagen interpretation, an interpreta-

tion of quantum mechanics which includes the Born statistical pic-ture of the wave function and which was propounded by Bohr.

God knows I am no friend of proba-

bility theory, I have hated it from the

first moment when our dear friendMax Born gave it birth. For it

could be seen how easy and simple

it made everything, in principle, ev-

erything ironed and the true prob-

lems concealed ...And actually not ayear passed before it became an offi-

cial credo, and it still is.

Schrodinger


11/13

More from Born:

If God has made the world a

perfect mechanism, He has at

least conceded so much to our

imperfect intellect that in order

to predict little parts of it, weneed not solve innumerable dif-

ferential equations, but can use

dice with fair success.

(Hence Einsteins famous comment, God does not play dice.)


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Bohm and indeterminacy

Bohm argued that:

[The] uncertain and incomplete

character of knowledge that exper-

iment at its present stage gives

us about what really happens inmicrophysics is the result of a real

indeterminacy is not in any way

justified.

He believed that the indeterminacy of QM results from our lack of

knowledge, and that eventually we might understand the deeper

level of reality that would allow exact predictions.


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The shut-up-and-calculate interpretation

Or we can take the Machian kind of world view and just say:

Quantum Mechanics is a man-made

generalization conveniently formulated

to account for various experimental ob-servations.

Quantum mechanics works! So lets just accept it and worry about

the deep meaning when weve got nothing better to do.

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