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Chapter 3.3a Microscopic World

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the topic of this video is development

of quantum theory behavior in the

microscopic world

the learning objective for this video is

on your screen

the physicist louis

wanted to know if electromagnetic radiation

radiation

which we classically thought of as a wave

wave

could at times depending on the

measurement and the context

have particle-like properties which the

particle being a photon

in the photoelectric effect for example if

if if that happens for electromagnetic

radiation what about for matter

what if matter which we thought of

classically as having particle like

uh properties could at times depending

on the context

and experiment have wave-like properties

particles have

wave like properties

so if if a particle such as an electron

does have wave-like properties

here's how we could think about it

instead of a particle

orbiting around a nucleus

at a fixed energy state you know the

ground state or it could be excited up to

an excited state but instead of a particle

particle

orbiting around the nucleus

instead there'd be a standing wave okay

so it's no longer a particle but actually

actually

uh the electron is a standing wave

and we can define a wavelength here so

this is what we refer to as

the debris wavelength and there's some

average distance from the nucleus which

would be the

radius here okay so what do i mean by

the debris wavelength well this is um

and it's a mathematical expression

where we have lambda which is the roy wavelength

wavelength

is equal to planck's constant over mass

times velocity so this is a v here it's

not we're not talking about frequency

um and uh

so here this is the de bruy wavelength

radiation um and so

again here m is mass v

and something important here is that we

can still

use aspects of the bohr model where

there are discrete energy states

available to

an electron except now we don't have to

think about it as a particle we can

think about the electron as a standing wave

wave

that whose radius can expand from

the center of uh of the atom or the nucleus

nucleus

so a mathematical expression here is

2 pi r r being the radius up above

is equal to n some integer n

multiplied by the derivative wavelength

and here n is equal to

1 2 3 or pretty much any

integer positive integer up to infinity

so you could see here that at a fixed

debris wavelength

all we have to do is increase this

integer value

n and that would predict we would

predict that

we would have a higher or a larger

radius of the standing wave okay so this

is reminiscent of of the bohr

model um okay so

if the prediction if that was the big

question then is there any evidence that

electrons do

have wave-like properties so this was tested

tested

by using a double slit style experiment

and again a double slit is one in which

waves passing through a double slit

will constructively and destructively

interfere on the other side of the slit

over in this region here that i'm

circling in red

and any detector here will pick up that

constructive interference where the waves

waves

added up to each other to produce a

higher signal and then the dark regions

here where there is destructive

interference where the waves

um uh cancel each other out

and so this wave pattern is definitely

or this interference pattern

uh is reminiscent of of of

wave-like behavior now if you

in the this experiment they took an

electron source and

they thought you know okay these are

particles that were blasting

um at this double slit but what they

ended up observing in detectors is that

if you let enough

time pass and you collect enough data here

here

what you'll see is an interference

pattern will eventually emerge and this

interference pattern

here we can say electrons do

have wave like

properties so this is this is a huge

you know uh development because now we

think electrons

don't have to be uh this particle

orbiting around a nucleus so atoms

aren't we don't have to think about them

as little solar systems of particles

zipping around

but instead for the electron we can

in the atom so now what i want to do

is go to a practice problem

where we can use the equation that i

showed above for the debris wavelength

so so here we can actually calculate the

the the wavelength

of a soft ball okay so this is a

macroscopic object

with a mass of 100 grams traveling at a

velocity of 35 meters per second

okay so we're making the assumption it

can be modeled as a single particle

so let's go ahead and do this um the

equation from above is the debris

wavelength is equal to planck's constant over

over

mass times velocity uh before i

planck's constant here i want to point

out to you that

planck's constant units are joules times seconds

seconds

but in this problem we have a mass on

the denominator of

and it's given to us in grams um and we

also have a velocity so meters and

seconds so there's no way that these units

units

can can be canceled as is but

a joule is also is equivalent to the

following units and it's kind of bizarre

kilogram meters squared per second squared

squared

that uh whole

ensemble of units kilograms times meter

squared divided by second squared is

actually equivalent to one joule

so i'm actually going to switch and use

um that in place of joules and so i'm

but i still have to tack on the seconds

here because that's also part of

planck's constant

okay so we're not going to use this

anymore we are going to use

this version over here but we can

simplify it a bit seconds here will

cancel out with the second squared on

the bottom

so when i write out planck's constant

over here

6.626 times 10 to the negative 34th

all i'm going to write is kilogram times

meter squared

times reciprocal seconds one over seconds

seconds

now this will make the units uh easier

to to cancel out

uh but we still have something else to

consider so the mass is 100 grams but

we have grams in the denominator

kilograms in the numerator so we need a

unit conversion here

so we know that there are 1000 grams per

one kilogram

and then the velocity units are fine uh

they are meter

per second so what we'll do here is uh

check that all of our units will cancel

we have kilograms we'll cancel kilograms

grams with grams meters with the meters

squared here

and reciprocal seconds with reciprocal

seconds so we'll be left with

units of meters which is exactly what we

would expect for

a wavelength and this should give you a

value of

1.9 times 10 to the negative 34

meters and so the the key

point here is we can calculate a

wavelength for a softball but

but the magnitude is so small that it

has no real

perceptible effect in the macroscopic

uh domain okay so there's no real

uh way to test whether a softball

has this wavelength it's so small but if

you have a really really really small

piece of matter such as an electron

and you calculate the debris wavelength

of that electron you'll find that it

actually has

a a wavelength that is larger

than the size of the particle itself and so

then in the macroscopic domain the

wave-like characteristics of matter become

become

one final thing here is the heisenberg

uncertainty principle

and this will come up when you take

a future chemistry class on quantum chemistry

chemistry

but here it is worth pointing out um

that it is fundamentally

impossible to determine

simultaneously and exactly

both the momentum

and the position of a particle

so what does that mean it means that

uh if particles have wave-like behaviors

there is some limit

into how well we can measure certain

properties of

of these particles and so briefly

what we can say here is that delta x is

the uncertainty

in a particle's position

and this will be um in the other unit

here this will be mass times the change

in velocity so this is

a change in momentum okay uh

the both of these multiplied by each

other are always

greater than or equal to this constant

planck's constant over

4 pi so what that means is

if your uncertainty

in the position of a particle goes

down so you become more certain of where

that particle is

you become less certain of its

momentum on the right and vice versa

so this is essentially the basis of the

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YouTube Transcript:
Chapter 3.3a Microscopic World