Why do superpositions occur? What causes them to occur?

[Hazel Appraiasal]

TL;DR Summary

I am a high schooler reading about superpositions. I cannot fully grasp the concept yet because I think I am just not getting some of the mechanics behind it. It sounds something like you'll get on paper, but when I really think hard about it, I am confused. I was also wondering if superpositions can occur on a much larger scale? Like Schrodinger's cat? It seems that they only occur in waves and such. Thank you for the help in advance.

https://en.wikipedia.org/wiki/Superposition_principle
https://isaacphysics.org/concepts/cp_superposition?stage=all
https://whatis.techtarget.com/definition/superposition

 

[EPR]

They are the natural state of matter and this is how the world works.

 

[PeroK]

Summary:: I am a high schooler reading about superpositions. I cannot fully grasp the concept yet because I think I am just not getting some of the mechanics behind it. It sounds something like you'll get on paper, but when I really think hard about it, I am confused. I was also wondering if superpositions can occur on a much larger scale? Like Schrodinger's cat? It seems that they only occur in waves and such. Thank you for the help in advance.

https://en.wikipedia.org/wiki/Superposition_principle
https://isaacphysics.org/concepts/cp_superposition?stage=all
https://whatis.techtarget.com/definition/superposition

There is a lot of misinformation about quantum superpositions online, so you need to be careful which sources you use.

The first point to note is that quantum states behave like vectors. In fact, they are vectors in an abstract sense. And vectors, fundamentally, allow the concept of superposition.

In QM, any system is described by its state (or state vector if you prefer). This is often denoted by $\psi$ or $|\psi \rangle$.

However, we also have the concept of energy eigenstates for a system and these are often written as $\psi_n$ or $|\psi_n \rangle$. These energy eigenstates play the role of basis vectors.

A system may be in an energy eigenstate, in which case we simply have: $\psi = \psi_5$ or $\psi = \psi_{16}$.

But, the system may instead be in a superposition of energy eigenstates. E.g., we could have: $$\psi = \psi_1 + \psi_2$$$$\psi = 2\psi_1 - \psi_4 + 3\psi_8$$
Note that the coefficients in front of each energy eigenstate give the relative weighing of that eigenstate in the superposition. This looks very like the more familiar vectors: $$\vec v = v_x \hat x + v_y \hat y + v_z \hat z$$ In other words, a quantum superposition of states is precsiely the same idea as a linear combination of vectors. In that sense, superpositions are really quite simple and certainly nothing new to QM.

But, the quantum mechanical nature of superpositions is seen when we measure the energy of a system in a superposition of energy eigenstates. The result of such a measurement is not a mixture of energies, but one definite energy. 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.

This might be a useful video to get you started on QM:

 

[Morbert]

In nature, it can be the case that we know as much as is physically possible to know about a system, but still have some uncertainty about what we will measure. Superposition helps us model this.

 

[Steve4Physics]

There is a lot of misinformation about quantum superpositions online, so you need to be careful which sources you use.

The first point to note is that quantum states behave like vectors. In fact, they are vectors in an abstract sense. And vectors, fundamentally, allow the concept of superposition.

In QM, any system is described by its state (or state vector if you prefer). This is often denoted by $\psi$ or $|\psi \rangle$.

However, we also have the concept of energy eigenstates for a system and these are often written as $\psi_n$ or $|\psi_n \rangle$. These energy eigenstates play the role of basis vectors.

A system may be in an energy eigenstate, in which case we simply have: $\psi = \psi_5$ or $\psi = \psi_{16}$.

But, the system may instead be in a superposition of energy eigenstates. E.g., we could have: $$\psi = \psi_1 + \psi_2$$$$\psi = 2\psi_1 - \psi_4 + 3\psi_8$$
Note that the coefficients in front of each energy eigenstate give the relative weighing of that eigenstate in the superposition. This looks very like the more familiar vectors: $$\vec v = v_x \hat x + v_y \hat y + v_z \hat z$$ In other words, a quantum superposition of states is precsiely the same idea as a linear combination of vectors. In that sense, superpositions are really quite simple and certainly nothing new to QM.

But, the quantum mechanical nature of superpositions is seen when we measure the energy of a system in a superposition of energy eigenstates. The result of such a measurement is not a mixture of energies, but one definite energy. 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.

This might be a useful video to get you started on QM:

What an excellent answer!

 

[Hazel Appraiasal]

There is a lot of misinformation about quantum superpositions online, so you need to be careful which sources you use.

The first point to note is that quantum states behave like vectors. In fact, they are vectors in an abstract sense. And vectors, fundamentally, allow the concept of superposition.

In QM, any system is described by its state (or state vector if you prefer). This is often denoted by $\psi$ or $|\psi \rangle$.

However, we also have the concept of energy eigenstates for a system and these are often written as $\psi_n$ or $|\psi_n \rangle$. These energy eigenstates play the role of basis vectors.

A system may be in an energy eigenstate, in which case we simply have: $\psi = \psi_5$ or $\psi = \psi_{16}$.

But, the system may instead be in a superposition of energy eigenstates. E.g., we could have: $$\psi = \psi_1 + \psi_2$$$$\psi = 2\psi_1 - \psi_4 + 3\psi_8$$
Note that the coefficients in front of each energy eigenstate give the relative weighing of that eigenstate in the superposition. This looks very like the more familiar vectors: $$\vec v = v_x \hat x + v_y \hat y + v_z \hat z$$ In other words, a quantum superposition of states is precsiely the same idea as a linear combination of vectors. In that sense, superpositions are really quite simple and certainly nothing new to QM.

But, the quantum mechanical nature of superpositions is seen when we measure the energy of a system in a superposition of energy eigenstates. The result of such a measurement is not a mixture of energies, but one definite energy. 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.

This might be a useful video to get you started on QM:

I see, can superpositions occur on a much larger scale then? Let's say, you were told to put the chicken out of the freezer last night, the next morning your mom leaves to go to work. You forget to put the chicken out of the freezer. Your mom comes home, is she in a superposition of being in all emotions until she looks at the chicken? Can superpositions occur at such scale? Thanks in advance

 

[romsofia]

PreoK's answer is very good, so read it a few times. He mentioned energy, but if you've taken chemistry before, you might remember Pauli's exclusion principle talking about electron spin. Particle spin is also formulated as a superposition before measurement.

But, a quick answer is: it's just math (for now). I'd stress really not getting bogged down quite yet into the philosophical debates on what superpositions mean pre-measurement (because it's a rabbit hole of fun IMO, and derails you from solving actual problems).

But, a more intuitive introduction to the concept of superposition is found in electromagnetism IMO.

A nice 3 minute video can be found here:

 

[StevieTNZ]

I see, can superpositions occur on a much larger scale then? Let's say, you were told to put the chicken out of the freezer last night, the next morning your mom leaves to go to work. You forget to put the chicken out of the freezer. Your mom comes home, is she in a superposition of being in all emotions until she looks at the chicken? Can superpositions occur at such scale? Thanks in advance

Macroscopic superposition is being tested by many groups - for example, one plans to attempt to place a (I think) 40kg mirror into a position superposition. It is also a nice time to test for things such as gravity potentially causing collapse of the wave function and the Leggett-Garg inequality (pertaining to macrorealism).

 

[PeterDonis]

Your mom comes home, is she in a superposition of being in all emotions until she looks at the chicken? Can superpositions occur at such scale?

At the "B" level, the simple answer to this question is no: we have no evidence that quantum superpositions occur at such a scale. The state your mom is in before she looks in the freezer is not a quantum superposition, it's a simple state of classical ignorance. Basically, while she doesn't yet know whether or not the chicken is frozen, there is no possible quantum interference between the two alternatives, "chicken frozen" and "chicken not frozen". It is the possibility of quantum interference between alternatives that makes a state a quantum superposition, instead of just a state of classical ignorance.

At more advanced levels, one could say it is in principle still an open question whether or not one can construct some kind of complicated experimental setup that would put a macroscopic object, possibly even a person, into a state that was truly a quantum superposition. The theory, taken literally, says it should be possible in principle, but the practical difficulties are immense and most physicists would probably say they will never be overcome. But in any case, any such scenario would be very different from an everyday scenario like not knowing whether the chicken was taken out of the freezer or not. So even taking into account the more advanced theoretical issues, what I said above is still valid.

 

[PeterDonis]

Macroscopic superposition is being tested by many groups

Note that all of these are very special experimental situations, and as I said in my previous post just now, they are very different from everyday occurrences where someone does not yet know the state of something (like whether or not the chicken is unfrozen).

 

[PeroK]

I see, can superpositions occur on a much larger scale then? Let's say, you were told to put the chicken out of the freezer last night, the next morning your mom leaves to go to work. You forget to put the chicken out of the freezer. Your mom comes home, is she in a superposition of being in all emotions until she looks at the chicken? Can superpositions occur at such scale? Thanks in advance

That's not a superposition of states. The chicken has either been taken out of the freezer or not. It's definitely one or the other.

A QM superposition is fundamentally different. The system is neither in one state nor the other. It's only when you measure the system that you get a definite value. It is not the case that the system always was in one of the eigenstates.

The state can be described by a wavefunction and this is called wavefunction collapse. Put simply the act of measuring the energy collapses the state of the system from a superposition to a single energy eigenstate corresponding to the energy measurement.

It's a mistake when you learn QM to imagine that complex macroscopic systems behave like elementary microscopic systems. The reason that QM took so long to discover is that systems of trillions of quantum particles loses all the fundamental characteristic QM behaviour, which all gets averaged out to what we know as classical behaviour.

A chicken comprising trillions of particles is fundamentally different from a single atom.

 

[Hazel Appraiasal]

Why

That's not a superposition of states. The chicken has either been taken out of the freezer or not. It's definitely one or the other.

A QM superposition is fundamentally different. The system is neither in one state nor the other. It's only when you measure the system that you get a definite value. It is not the case that the system always was in one of the eigenstates.

The state can be described by a wavefunction and this is called wavefunction collapse. Put simply the act of measuring the energy collapses the state of the system from a superposition to a single energy eigenstate corresponding to the energy measurement.

It's a mistake when you learn QM to imagine that complex macroscopic systems behave like elementary microscopic systems. The reason that QM took so long to discover is that systems of trillions of quantum particles loses all the fundamental characteristic QM behaviour, which all gets averaged out to what we know as classical behaviour.

A chicken comprising trillions of particles is fundamentally different from a single atom.

Why do they occur though? Is there a phenomena that causes superpositions to occur? Or am I misunderstanding and you've already stated this before and I just have to back read it?

 

[PeroK]

Why do they occur though? Is there a phenomena that causes superpositions to occur? Or am I misunderstanding and you've already stated this before and I just have to back read it?

This is why you need to start learning about QM systematically. One key concept is that not all measurables are compatible. You may have heard of the HUP (Heisenberg Uncertainty Principle). This states that position and momentum are incompatible observables. One consequence of this is that if you measure the momentum of a particle, then the state collapses to an eigenstate of momentum, but this is a superposition of position eigenstates. And, vice versa.

In that sense, a free particle is always in a superposition of momentum and/or position eigenstates.

You may also have heard that in the hydrogen atom, the electron is a "cloud of probability" around the nucleus. More precisely, if the hydrogen atom is in the ground state (or any energy eigenstate), then the electron is in a superposition of position eigenstates - which is one way to describe the wavefunction.

And, if you did measure the position of the electron, that would throw the atom into a superposition of energy eigenstates.

In this sense, quantum systems are always in a superposition of one description or other. You can't avoid superpositions. Again, this is directly analagous to a vector:

The vector $\vec v = \hat x + \hat y$ is in a superposition of the basis vectors $\hat x$ and $\hat y$. But, if we define a new basis vector $\vec e = \hat x + \hat y$, then $\vec v = \vec e$ appears no longer to be in a superposition. The question is not whether the vector $\vec v$ is in a superposition, as it is always both a single, well-defined vector and a superposition of vectors at the same time.

 

[DennisN]

I see, can superpositions occur on a much larger scale then?

Here is an example of such an experiment (though it's not as big as a cat ):

Article: Quantum effect spotted in a visible object (PhysicsWorld)
(note that this is just an article about the experiment, and some quotes in it may not be universally agreed on)

Paper: A. D. O’Connell et al, Quantum ground state and single-phonon control of a mechanical resonator (Nature volume 464, pages 697–703 (01 April 2010))

In addition to what already has been said above about superposition, here's a short video on Sixty Symbols about Schrödinger's Cat (which is a thought experiment) and superposition: Schrödinger's Cat - Sixty Symbols.

 

[PeroK]

Here is an example of such an experiment (though it's not as big as a cat ):

Article: Quantum effect spotted in a visible object (PhysicsWorld)

That article contains the sort of misinformation about QM that I warned about in post #2.

 

[DennisN]

That article contains the sort of misinformation about QM that I warned about in post #2.

Yes, I am aware of that. So I added ("note that this is just an article about the experiment, and some quotes in it may not be universally agreed on").

 

[PeroK]

Yes, I am aware of that. So I added ("note that this is just an article about the experiment, and some quotes in it may not be universally agreed on").

I suggest we encourage the OP to start reading about QM systematically, rather than dip into a confusion of ideas such as those presented in the the sixty symbols video. That's like dipping into a box of chocolates, rather than eating a proper meal.

 

[Hazel Appraiasal]

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. Hoping to know as much QM as you do someday. Cheers!

Also, I hope you don't mind me asking a lot of questions, I'm really interested in this.

I suggest we encourage the OP to start reading about QM systematically, rather than dip into a confusion of ideas such as those presented in the the sixty symbols video. That's like dipping into a box of chocolates, rather than eating a proper meal.

 

[vanhees71]

I think a good starting point is the qm volume of Susskind's "Theoretical Minimum Series":

https://www.amazon.com/dp/0465062903/?tag=pfamazon01-20

 

[PeroK]

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. Hoping to know as much QM as you do someday. Cheers!

Also, I hope you don't mind me asking a lot of questions, I'm really interested in this.

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

 

[Nugatory]

Also, I hope you don't mind me asking a lot of questions, I'm really interested in this.

The Susskind book recommended above by @vanhees71 would be a really good start. If you find that it demands more math than you can handle ("I am a high schooler..." covers a lot of ground) there is also Giancarlo Ghirardi's book "Sneaking a look at god's cards" which is more layman-friendly.

Ghirardi is not a substitute for Susskind or a real textbook explaining the real thing using the language of physics, which is math. However it is far better than most of the random stuff floating around on the internet, or trying to make sense of often good but always unstructured wikipedia articles.

 

[Lynch101]

Why do they occur though? Is there a phenomena that causes superpositions to occur? Or am I misunderstanding and you've already stated this before and I just have to back read it?

'When you use the term 'superposition' what is it that you have in mind? Are you talking about the mathematical idea of a 'superposition' or are you talking about what happens in the lab, during an experiment?

If you're talking about what happens in the lab then there is no universal agreement about what the 'superposition' is. Some people will say that the quantum system exists in many places or states at once, while others will say that a 'superposition' is just a mathematical tool to help make predictions about experiments. Others will say something different still.

However, in the lab, there is a piece of equipment which prepares the quantum system to be used in the experiment. It is this which puts the system into a 'superposition', whatever that may turn out to be.

If you apply your curiosity to learning the mathematics of quantum theory you might be the one to solve the mystery

 

[vanhees71]

The most important thing is to note that it doesn't make sense to say "a state is a superposition" without telling "superposition" wrt. which basis.

If the system is prepared in a pure state, this state can be represented by a normalized vector $|\psi \rangle$ in a Hilbert space (a vector space over the complex numbers as scalars with a scalar product, denoted by $\langle \phi|\psi \rangle$).

Now if you measure some quantity, e.g., a spin component, then the you have to use the eigenvectors of the corresponding operator, representing the measured observable, to calculate the probability of the outcome of the measurement, which is always an eigenvalue of the operator. So the only meaning of the quantum state is to provide the probability for the outcome of a measurement. If $|\psi \rangle$ is a spin state of some particle is prepared (e.g., some neutral atom with a Stern-Gerlach experiment), then the probability to find the possible values $\sigma_z \in \{\pm \hbar/2 \}$ is given by
$$P(\sigma_z)=|\langle \sigma_z|\psi \rangle|^2.$$
On the other hand you can expand the state uniquely in terms of $\sigma_z$-eigenvectors,
$$|\psi \rangle=\psi_{1/2} |1/2 \rangle + \psi_{-1/2} |-1/2 \rangle.$$
Then the probabilities are
$$P(\hbar/2)=|\psi_{1/2}|^2, \quad P(\hbar/2)=|\psi_{-1/2}|^2.$$
So the particle has a definite value of $\sigma_z$ if and only if $|\psi_{1/2}|^2=1$ (and then of course you must have $|\psi_{-1/2}|^2=0$) or $|\psi_{-1/2}|^2=1$ (and then of course you must have $|\psi_{1/2}|^2=0$).

Now if both $|\psi_{\pm 1/2}|^2 \neq 0$ (but with $|\psi_{1/2}|^2+|\psi_{-1/2}|^2=1$, becaue $|\psi \rangle$ must be normalized to 1) one sloppily says "|\psi \rangle is in a superposition of $\sigma_z$ eigenstates", which implies that $\sigma_z$ has not a determined value through the preparation of the particle in the spin state $|\psi \rangle$.

The quantum formalism of spin-1/2 particles however tells you that $|\psi \rangle$ is also an eigenstate of a spin component in another direction than the $z$ direction, $\vec{n}$, i.e., $\sigma_{\vec{n}}$ then takes a determined value ($+\hbar/2$ or $-\hbar/2$). Wrt. to this observable the same state then is "not in a superposition".

 

[Hazel Appraiasal]

'When you use the term 'superposition' what is it that you have in mind? Are you talking about the mathematical idea of a 'superposition' or are you talking about what happens in the lab, during an experiment?

If you're talking about what happens in the lab then there is no universal agreement about what the 'superposition' is. Some people will say that the quantum system exists in many places or states at once, while others will say that a 'superposition' is just a mathematical tool to help make predictions about experiments. Others will say something different still.

However, in the lab, there is a piece of equipment which prepares the quantum system to be used in the experiment. It is this which puts the system into a 'superposition', whatever that may turn out to be.

If you apply your curiosity to learning the mathematics of quantum theory you might be the one to solve the mystery

Hmm, what about electron superposition spins? What kind of superposition are they?

 

[DennisN]

I suggest we encourage the OP to start reading about QM systematically, rather than dip into a confusion of ideas such as those presented in the the sixty symbols video. That's like dipping into a box of chocolates, rather than eating a proper meal.

What's good about the Sixty Symbols video, in my opinion, is that it makes very clear that Schrödinger's cat is a thought experiment, and it also describes why the thought experiment was made in the first place. It seems the OP wondered about that in the first post (like so many have done before).

 

[Hazel Appraiasal]

What's good about the Sixty Symbols video, in my opinion, is that it makes very clear that Schrödinger's cat is a thought experiment, and it also describes why the thought experiment was made in the first place. It seems the OP wondered about that in the first post (like so many have done before).

I've read about how it was a thought experiment, so I am caught up with that. I was just wondering how superpositions happen lol. Thanks for the help though

 

[PeroK]

I've read about how it was a thought experiment, so I am caught up with that. I was just wondering how superpositions happen lol. Thanks for the help though

Note that Schrodinger's cat is a thought experiment designed to show that QM can't make complete sense. It's not a good place to start. In particular, it doesn't help you understand QM at all. Quite the reverse, you need an understanding of QM to think seriously about what this thought experiment is saying.

This is another difference between popular QM and undergraduate QM. In the latter, you will learn QM first and then start thinking about the deeper questions. In popular science, you will tend to jump in with the deeper questions first. The problem is (and we see it a lot on this site) that unless you thoroughly understand what QM is saying, it's difficult to tackle these paradoxical scenarios. That leads to the risk (and we see this a lot as well), that someone may give up on QM before they've even begun learning about it.

 

[Nugatory]

Hmm, what about electron superposition spins? What kind of superposition are they?

They are a superposition of other spin states. For example, the state that we call "spin up" is also a superposition of the spin states "spin left" and "spin right", or of any other directions we choose, but it's the same state no matter how we talk about it.

It's really no more mysterious than being able to write the same number as 5 or 3+2 or 4+1, a kind of superposition that we do all the time so reflexively that we often aren't aware if it: when we have five people and four chairs we naturally use the fact that the number 5 is also the sum of the numbers 4 and 1 to conclude that we need one more chair. Quantum superpositions are a bit more complicated because (as as been said above) quantum states add like vectors instead of numbers, but the basic notion is still that a state can be written as the sum of other states.

 

[Lynch101]

Hmm, what about electron superposition spins? What kind of superposition are they?

Are you asking about the mathematical formulation of it or the physical experimental set-up?

 

[Hazel Appraiasal]

Note that Schrodinger's cat is a thought experiment designed to show that QM can't make complete sense. It's not a good place to start. In particular, it doesn't help you understand QM at all. Quite the reverse, you need an understanding of QM to think seriously about what this thought experiment is saying.

This is another difference between popular QM and undergraduate QM. In the latter, you will learn QM first and then start thinking about the deeper questions. In popular science, you will tend to jump in with the deeper questions first. The problem is (and we see it a lot on this site) that unless you thoroughly understand what QM is saying, it's difficult to tackle these paradoxical scenarios. That leads to the risk (and we see this a lot as well), that someone may give up on QM before they've even begun learning about it.

What is an eigenstate?

 

[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