Problem Solving Strategies

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John Rennie

06:36

@AarushiAgarwal Hi :-)

I'm not familiar with "color and hardness boxes"

Aarushi Agarwal

ohh

well i'll send u link to this lecture

ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/…

the color and hardness are actually the spin property of the electron , but for simplification ambiguous names are taken

John Rennie

Ah, OK, they are just used as analogies.

Superposition is actually very simple mathematically, but it involves ideas that are unintuitive and can be hard to get your head around.

I can try and explain it using the example of waves on a guitar string if you want.

Aarushi Agarwal

06:59

sure

would be helpful

John Rennie

07:10

@AarushiAgarwal Have you learned about standing waves?

Kavin Ishwaran

@JohnRennie I have a doubt about the Standing waves, should the two waves exactly coincide to produce a standing wave ?

Or its a different thing ?

John Rennie

@KavinIshwaran If we are talking about a guitar string then the amplitude at the ends has to be zero because the string is clamped at the ends. Yes?

Kavin Ishwaran

@JohnRennie yes

John Rennie

So the standing wave has to look something like this:

Yes ... ?

Kavin Ishwaran

yes

John Rennie

07:14

That can only happen if we have two identical waves, one moving left and one moving right. Then when we add the waves we find they always cancel at the ends to give a zero displacement.

Kavin Ishwaran

@JohnRennie yes but when seeing that visually, I always wonder if the two waves must exactly coincide to make a standing waves

John Rennie

A wave moving in the positive x direction (i.e. moving right) has the equation y(x,t) = sin(ωt - kx)

OK so far?

Kavin Ishwaran

yes

@JohnRennie we are taking this in a two dimensional graph ?

Aarushi Agarwal

@JohnRennie yeah

John Rennie

Yes, the amplitude depends on both distance and time. In my animation the amplitude changes as we move along the x axis, and it also changes in time. Yes?

Aarushi Agarwal

07:18

yes

John Rennie

@AarushiAgarwal OK, so if we consider a standing wave on a guitar string it looks like the diagram above:

6 mins ago, by John Rennie

OK so far?

Aarushi Agarwal

yes

John Rennie

But this is just the fundamental oscillation. We could also have the guitar string vibrating in the first harmonic:

Yes?

Aarushi Agarwal

yes

John Rennie

But, we can add these two vibrations together and this is a perfectly reasonable way for the string to oscillate. If we add together these two waves we get:

Aarushi Agarwal

07:21

i understand the superposition of waves

but not in the superposition in case for the analogy i told yo uabout

John Rennie

@AarushiAgarwal In that case you understand superposition because that is all that happens in quantum mechanics.

Kavin Ishwaran

@JohnRennie So it depends on the distance between the two waves, it is not necessary to get coincided ?

John Rennie

@AarushiAgarwal Suppose you have an electron in a hydrogen atom, then it behaves a bit like a wave. The fundamental is the 1s and the first harmonic is the 2s.

OK so far?

@KavinIshwaran Let me finish answering Aarushi and we can come back to this ...

Kavin Ishwaran

@JohnRennie Sure :-)

John Rennie

@AarushiAgarwal OK so far?

Aarushi Agarwal

07:26

@JohnRennie ok

John Rennie

But the electron can also be in a state that is a mixture of 1s and 2s. Then we say it is in a superposition of the two states.

Aarushi Agarwal

it seems unintuitive like u said

@JohnRennie how ?

John Rennie

It turns out this superposition isn't stable and it either absorbs a photon and goes to the 2s or emits a photon and goes to the 1s. In fact this is exactly how a hydrogen atom absorbs and emits photons.

@AarushiAgarwal The hydrogen atom has to obey the Schrodinger equation, and the 1s, 2s, 2p, etc states are just the solutions to the Schrodinger equation for a hydrogen atom.

But it turns out the state 1s + 2s is also a solution. In fact any sum of states is a solution, and that means the hydrogen atom can be in a state that is a mixture of any of the 1s, 2s, 2p, etc states.

And it really can. As I said above this is how hydrogen atoms absorb and emit photons to change their state so this is really true.

The thing is that superposition seems obvious when it's waves on a guitar string, but unintuitive when it's states of a hydrogen atom, but really there is no difference.

Aarushi Agarwal

yeh

thanks

John Rennie

Where it gets really hard to get your head around is that this applies to all proproties of the electron.

That's what the article you linked is trying to say.

Electrons obviously don't have a colour, but if they did then they could be in superpositions of different colours.

Or another example is that electrons are always in a superposition of different positions.

Kavin Ishwaran

07:36

@JohnRennie when saying this, this implies that an electron can be in all energy states. So a photon can also be superposed ?

John Rennie

Yes, all photons are in a superposition of different energies i.e. the photon always has a range of wavelengths/frequencies.

Kavin Ishwaran

@JohnRennie "So it depends on the distance between the two waves, it is not necessary to get coincided ?"

John Rennie

OK, back to standing waves:

Suppose we add together the right and left moving waves then we get:

y(x,t) = sin(ωt - kx) + sin(ωt + kx)

Yes?

Kavin Ishwaran

Sorry sir, got an important call

@JohnRennie yes

John Rennie

I'm busy for a few moments ...

Kavin Ishwaran

07:52

Ok, we will see when you are free :-)

John Rennie

@KavinIshwaran Hi

Kavin Ishwaran

Hi !

John Rennie

OK, there is an identity for sin(a) + sin(b) = 2 sin( (a+b)/2 ) cos( (a-b)/2 )

Yes?

Kavin Ishwaran

yes

John Rennie

So we can use this to simplify our expression, and we get:

y(x,t) = sin(ωt - kx) + sin(ωt + kx) = 2 sin(ωt) cos(-kx)

Yes?

Kavin Ishwaran

07:59

yes

John Rennie

And this is just a standing wave with the nodes at kx = ±π/2

So we don't have to do anything special to create the standing wave. We just add the two travelling waves.

Kavin Ishwaran

so the distance between them doesnt matter

just add two parallelly travelling waves

John Rennie

They don't have a "distance" between them. They are both infinite waves i.e. extending from x = -∞ to +∞.

Kavin Ishwaran

@JohnRennie ok then

got it

Thank you for some clarification sir :-)

John Rennie

You can add a phase offset: y(x,t) = sin(ωt - kx) + sin(ωt + kx + φ)

For some phase offset φ

All that does is shift the nodes along the x axis.

Kavin Ishwaran

08:03

I see

2 hours later…

Aarushi Agarwal

10:12

@JohnRennie if photons are in a superposed state of energies , how is the amount discrete ?

John Rennie

@AarushiAgarwal We normally say the energy of a photon is E = hf. Yes?

John Rennie

10:24

Now, because the frequency isn't perfectly defined we should write it as f ± Δf where Δf is the amount the frequency is spread out, and plugging this into our equation for the energy we find we get an energy E+ ΔE where ΔE = hΔf.

But in most cases Δf is much smaller than f, so ΔE << E.

So for example when we say the 2p ⟶ 1s transition in a hydrogen atom emits a photon with energy 10.2eV, we really mean it's 10.2eV ± a small amount. So the energy is still discrete in the sense that it is still 10.2ev, and couldn't be 9.2 or 11.2eV because ΔE is so small.

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