2.3.2_slides linearity and normalization

8/10/2019 2.3.2_slides Linearity and Normalization

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2.3 Particle in a box

Slides: Video 2.3.2 Linearity and

normalizationText reference: Quantum Mechanics

for Scientists and EngineersSection 2.4 – 2.5



The particle in a box

Linearity and normalizatio

Quantum mechanics for scientists and engineers Da



Linearity and Schrödinger’s equation

We see that Schrödinger’s equation is linear

The wavefunction appears only in first ordthere are no second or higher order term

such as 2

or 3

So, if is a solution, so also is a

this just corresponds to multiplying bothsides by the constant a

2

2

2V E

m

r



Normalization of the wavefunction

Born postulated

the probability of finding a particle

near a point r is

Specifically let us define as a

“probability density”For some very small (infinitesimal)

volume d 3

r around rthe probability of finding the particl

in that volume is

P r

2

r

P r

3P d r r




The sum of all such probabilities should be

So

Can we choose so that we can use

as the probability densitynot just proportional to probability den

Unless we have been luckyour solution to Schrödinger’s equdid not give a so that

3

1P d r r

r

r

2 3 1d r r




Generally, this integral would give some othreal positive number

which we could write aswhere a is some (possibly complex)

numberThat is,

But we know that if is a solution ofSchrödinger’s equation

so also is

21/ a

2 3

2

1d

a r r

r

a r




So

if we use the solution instead of

then

and we can use as the probabildensity, i.e.,

would then be called a

“normalized wavefunction”

N a

2 3 1

N d r r

2

N P r r

2

N r

N r




So, to summarize normalization

we take the solution we have obtained

from Schrödinger’s wave equationwe integrate to get a number we

callthen we obtain the normalized

wavefunction for which

and we can use as theprobability density

2

r2

1/ a

N a

2 3 1

N d r r

2

N r



Technical notes on normalization

Note that normalization only sets themagnitude of a

not the phasewe are free to choose any phase for a

or indeed for the original solution a phase factor is justanother number by which we camultiply the solution

and still have a solution

exp i




If we think of space as infinite

functions like , , and

cannot be normalized in this wayTechnically, their squared modulus is

not “Lebesgue integrable”They are not “L2” functions

This difficulty is mathematical, not physicalIt is caused by over-idealizing themathematics to get functions that are

simple to use

sin kx cos kz exp i k




There are “work-arounds” for this difficulty

1 - only work with finite volumes in actua

problemsthis is the most common solution

2 - use “normalization to a delta functionintroduces another infinity to

compensate for the first one

This can be done

but we will try to avoid it

2.3.2_slides linearity and normalization

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