High School Why do superpositions occur? What causes them to occur?

[Hazel Appraiasal]

Here is a more organised version of what I've been saying about a superposition of eigenstates. Can you makes sense of this?

http://physics.gmu.edu/~dmaria/590 Web Page/public_html/qm_topics/superposition/superposition.html

I am so sorry to say this but I can't make sense of this. Hahahaha
I get the first part where you are getting the sum of the probability of the states but after that, it kinda just falls apart for me.

I'm so sorry

 

[PeroK]

I am so sorry to say this but I can't make sense of this. Hahahaha
I get the first part where you are getting the sum of the probability of the states but after that, it kinda just falls apart for me.

I'm so sorry

QM is not an elementary subject. If you want to learn it, then that is an incentive to study classical physics and the mathematics that goes with it.

 

[Hazel Appraiasal]

We always get one of $E_1, E_2, E_3 \dots$. The square of the coefficients tells us the probability that a measurement returns that energy value. In our example above, the relative probability of getting $E_1, E_4$ or $E_8$ from a measurement of the state $\psi = 2\psi_1 - \psi_4 + 3\psi_8$ would be $4, 1$ and $9$. In other words, on average out of $14$ measurements of energy in a system in that state, we would get $E_1$ four times, $E_4$ once and $E_8$ nine times.
simhBM[/MEDIA]

What does E mean here?

 

[PeroK]

What does E mean here?

Energy!

 

[Hazel Appraiasal]

H

Energy!

Hmm, so let me get this straight

In QM, a state of a quantum system is denoted by the symbol "psi" or $\psi$ or $|\psi \rangle$.

A quantum state can be in a state where they're at that point it just denoted by $\psi = \psi_5$ or $\psi = \psi_{16}$ for example (I don't know if the importance of the numbers at the bottom)

But when it is in a superposition of states it, for example, could be in

$$\psi = \psi_1 + \psi_2$$
$$\psi = 2\psi_1 - \psi_4 + 3\psi_8$$

And the numbers next to the psi symbols are the possibilities you will get that state just like in your example

$\psi = 2\psi_1 - \psi_4 + 3\psi_8$ would be $4, 1$ and $9$. In other words, on average out of $14$ measurements of energy in a system in that state, we would get $E_1$ four times, $E_4$ once and $E_8$ nine times.

Correct me if I am wrong but this is what I'm getting

 

[PeroK]

H

Hmm, so let me get this straight

In QM, a state of a quantum system is denoted by the symbol "psi" or $\psi$ or $|\psi \rangle$.

A quantum state can be in a state where they're at that point it just denoted by $\psi = \psi_5$ or $\psi = \psi_{16}$ for example (I don't know if the importance of the numbers at the bottom)

But when it is in a superposition of states it, for example, could be in

$$\psi = \psi_1 + \psi_2$$
$$\psi = 2\psi_1 - \psi_4 + 3\psi_8$$

And the numbers next to the psi symbols are the possibilities you will get that state just like in your example

$\psi = 2\psi_1 - \psi_4 + 3\psi_8$ would be $4, 1$ and $9$. In other words, on average out of $14$ measurements of energy in a system in that state, we would get $E_1$ four times, $E_4$ once and $E_8$ nine times.

Correct me if I am wrong but this is what I'm getting

Yes, a classical system would have a definite energy $E$. And, in general this energy could be any number.

Some quantum systems - like an atom - may only take certain energy values. There is a lowest possible energy $E_1$, then a next energy level $E_2$ etc. These are called energy eigenstates.

That property is called discreteness. I.e. energy can only take certain quantised values.

Quantum systems may also be in a superposition of many energy eigenstates. They do not have a definite energy until measured. That's a second characterictic that is not found in classical systems.

 

[Hazel Appraiasal]

Yes, a classical system would have a definite energy $E$. And, in general this energy could be any number.

Some quantum systems - like an atom - may only take certain energy values. There is a lowest possible energy $E_1$, then a next energy level $E_2$ etc. These are called energy eigenstates.

That property is called discreteness. I.e. energy can only take certain quantised values.

Quantum systems may also be in a superposition of many energy eigenstates. They do not have a definite energy until measured. That's a second characterictic that is not found in classical systems.

Okay okay, I am starting to get the hang of it. So does the thing I said above practically explain how superpositions happen? Or are there more things needed for them to occur? Like interference waves, wave function collapse and what not

 

[PeroK]

Okay okay, I am starting to get the hang of it. So does the thing I said above practically explain how superpositions happen? Or are there more things needed for them to occur? Like interference waves, wave function collapse and what not

In a deep sense superpositions don't occur, they are. Take the double-slit experiment. The particle starts in an infinite superposition of position eigenstates. The particle encounters the barrier with the double-slit and this superposition changes into two distinct superpositions which then interfere with each other. The critical thing is not whether the particle is in a superposition of states, but the nature of the superposition. The particle is always in a superposition of states. In the double-slit, the superposition of states after the slits is more intersting than the superposition before the slits.

Note that we usually talk about these superpositions as the wavefunction in this case.

This is why popular science accounts are not quite right when they claim that the particle acts like a particle before the slits and like a wave afterwards. The particle is always a particle. It has a simple wavefunction before the slits and a more interesting wavefunction after the slits. It's all quantum behaviour, driven by a wavefunction: some wavefunctions result in classical particle-like behaviour and some wavefunctions result in classical wave-like behaviour.

In summary, what happens to a particle determines its wavefunction, hence the particular superposition of states. But, it's always a superposition if you look at it the right way.

 

[Hazel Appraiasal]

In a deep sense superpositions don't occur, they are. Take the double-slit experiment. The particle starts in an infinite superposition of position eigenstates. The particle encounters the barrier with the double-slit and this superposition changes into two distinct superpositions which then interfere with each other. The critical thing is not whether the particle is in a superposition of states, but the nature of the superposition. The particle is always in a superposition of states. In the double-slit, the superposition of states after the slits is more intersting than the superposition before the slits.

Note that we usually talk about these superpositions as the wavefunction in this case.

This is why popular science accounts are not quite right when they claim that the particle acts like a particle before the slits and like a wave afterwards. The particle is always a particle. It has a simple wavefunction before the slits and a more interesting wavefunction after the slits. It's all quantum behaviour, driven by a wavefunction: some wavefunctions result in classical particle-like behaviour and some wavefunctions result in classical wave-like behaviour.

In summary, what happens to a particle determines its wavefunction, hence the particular superposition of states. But, it's always a superposition if you look at it the right way.

Hmm, so they don't occur, per se, they just happen that way?

Also, what does the number mean in the subscript?
$$\psi = 2\psi_1 - \psi_4 + 3\psi_8$$

 

[PeroK]

Hmm, so they don't occur, per se, they just happen that way?

Also, what does the number mean in the subscript?
$$\psi = 2\psi_1 - \psi_4 + 3\psi_8$$

$\psi_1$ is the energy eigenstate corresponding to energy $E_1$ etc.

 

[Hazel Appraiasal]

$\psi_1$ is the energy eigenstate corresponding to energy $E_1$ etc.

So, $\psi_1$ refers to a state that it could be in? But in the video example, you gave me, I didn't see a lot of it

 

[PeroK]

So, $\psi_1$ refers to a state that it could be in? But in the video example, you gave me, I didn't see a lot of it

He was looking at quantum spin, rather than energy eigenstates.

 

[Hazel Appraiasal]

He was looking at quantum spin, rather than energy eigenstates.

So, if I understand correctly

The difference between using regular "psi" and "ket" is that psi is used to denounce the state that something is in while ket refers to adding the possibility of the two states together?

 

[PeroK]

So, if I understand correctly

The difference between using regular "psi" and "ket" is that psi is used to denounce the state that something is in while ket refers to adding the possibility of the two states together?

Just different notation for the same thing.

 

[Hazel Appraiasal]

Just different notation for the same thing.

Okay okay, I guess these solves a lot for me

THANK YOU SO MUCH PEROK, I never would have gotten the lesson without you. Cheers Bro!

 

[Dali]

Thank you PeroK, the answers you have given me are quite much than what I expected so I am really grateful. I still don't know how superpositions happen, maybe I am just rushing to understand things. [...]

Hi Hazel! You have already gotten great answers, and seem to start to get a grip on the QM formalism of superpositions!

The question "why" anything occurs though is always a tricky slope with no definite answer... If you ask why A is true and someone says it is because of B, you just then transferred the question to why B is true, etc with no end. Nevertheless, I think you might benefit from understanding a bit more about the kind of experiments that forced us to conclude that classical physics totally failed to describe microscopical stuff, and that we *had* to introduce superpositions (and the whole formalism of quantum mechanics!) in order to not be wrong about what happens in nature. Studying those experiment in a little more detail will also give you a better understanding of what the concept of superpositions means in terms of actual observable facts.

In short (as also mentioned by @PeterDonis) it is the possibility of interference between (classically) different and mutually exclusive states that force us to introduce the concept of superpositions. The simplest example is the double slit experiment, where it is easy to show that if we make the (very plausible!) assumption that particles only can be at one position at any given time, the results we would get can never match the interference effects that we do see in real experiments. That is the very big difference between a superposition and classical ignorance. (Classical ignorance would be that particles have positions at all times but we sometimes just don't know which exactly where they are. That is called a "classical mixture" of states. Many popular scientific sources are not really clear about this crucial distinction!)

I always thought Richard Feynman has the best basic explanation of the double-slit experiment and superpositions (and a lot more!) in his excellent books "The Feynman Lectures of Physics". These are available free online nowadays, and I think you would enjoy the first chapter in Vol III: https://www.feynmanlectures.caltech.edu/III_01.html (in particular section 1-5)

 

[Hazel Appraiasal]

Hi Hazel! You have already gotten great answers, and seem to start to get a grip on the QM formalism of superpositions!

The question "why" anything occurs though is always a tricky slope with no definite answer... If you ask why A is true and someone says it is because of B, you just then transferred the question to why B is true, etc with no end. Nevertheless, I think you might benefit from understanding a bit more about the kind of experiments that forced us to conclude that classical physics totally failed to describe microscopical stuff, and that we *had* to introduce superpositions (and the whole formalism of quantum mechanics!) in order to not be wrong about what happens in nature. Studying those experiment in a little more detail will also give you a better understanding of what the concept of superpositions means in terms of actual observable facts.

In short (as also mentioned by @PeterDonis) it is the possibility of interference between (classically) different and mutually exclusive states that force us to introduce the concept of superpositions. The simplest example is the double slit experiment, where it is easy to show that if we make the (very plausible!) assumption that particles only can be at one position at any given time, the results we would get can never match the interference effects that we do see in real experiments. That is the very big difference between a superposition and classical ignorance. (Classical ignorance would be that particles have positions at all times but we sometimes just don't know which exactly where they are. That is called a "classical mixture" of states. Many popular scientific sources are not really clear about this crucial distinction!)

I always thought Richard Feynman has the best basic explanation of the double-slit experiment and superpositions (and a lot more!) in his excellent books "The Feynman Lectures of Physics". These are available free online nowadays, and I think you would enjoy the first chapter in Vol III: https://www.feynmanlectures.caltech.edu/III_01.html (in particular section 1-5)

Thanks for the advice bro, I really appreciate it! God bless my man