Bra-ket Notation: Explained in Layman's Terms + How to Notate

• JDude13
In summary, bras and kets exist in different spaces, and the notation for expressing the probability that a particle will be in a particular state is written as \Lambda_n = |\langle x_n|\Psi\rangle|^2.
JDude13
Could someone please explain bra-ket notation in layman's terms?
Also could you please tell me how to notate this (bra-ket or otherwise)?

The probability of $$x_n$$ is equal to $$\Lambda_n$$.

$$\Lambda_n$$ is a value between 0 and 1.

$$x_n$$ is, of course, position.

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Hi JDude13!

Technically, bras and kets exist in different spaces …

for example, you can consider them as row vectors and column vectors.

The pointy bit tells you which way round you should put them …

since matrices (operators) have straight sides, you can only put the straight side of the bra or ket next to a matrix.

And if you put the two pointy sides of a bra and ket together, the result has straight sides, so it's a matrix, but if instead you put the two straight sides together, the reuslt has pointy sides, and is a number.

That's all there is, really.

(oh, and it's P(xn = Λn))

$$P(x_n=\Lambda_n)$$?
But doesn't this say that the probability of event $$x_n$$ is the same as the probability of event $$\Lambda_n$$?
I wanted to express that the probability of event $$x_n$$ is the same as the value of $$\Lambda_n$$.

Would it maybe be $$P(x_n)=\Lambda_n$$?

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JDude13 said:
I wanted to express that the probability of event $$x_n$$ is the same as the value of $$\Lambda_n$$.

Would it maybe be $$P(x_n)=\Lambda_n$$?

I'm confused … you said …
JDude13 said:
$$x_n$$ is, of course, position.

How can you have a sentence starting "The probablity of position is …" ?

tiny-tim said:
How can you have a sentence starting "The probablity of position is …" ?

Okay... It's actually a developing theory so I don't want to go into details but...

I want $$\Lambda_n$$ to represent the probability of an elementary subatomic particle being at point $$x_n$$.

Then you'll need to give that position a name, say x, so that you can write P(x = xn) = Λn

Okay... So, in English, that means that the probability of $$x$$ being equal to $$x_n$$ is $$\Lambda_n$$. Right?
I just assumed that an equals sign couldn't be included inside a bracket.
We haven't done probability like this in school, yet and I never thought to research it independantly as it has never occurred in my study.

JDude13 said:
Right?

Right.
I just assumed that an equals sign couldn't be included inside a bracket.

We do it all the time!

JDude13 said:
Could someone please explain bra-ket notation in layman's terms?
Why would you want a non-mathematical explanation of something mathematical? I've seen many of these requests, and they don't make sense to me. Anyway, what you're asking about seems to be unrelated to bras and kets. (I would use the notation tiny-tim suggested in #6).

Hi Fredrik!

I just don't know how you could write that
Fredrik said:
… i.e. it's just a vector written in a funny way. …
… without a smilie!

Yes, I can see how you would feel that way.

The bra-ket notation for a state $$\Psi$$ to be in a position eigenstate $$x_n$$ is written as $$\Lambda_n = |\langle x_n|\Psi\rangle|^2$$. However, if you don't understand how bras and kets work at a mathematical level, that may or may not be of much use to you. I'd suggest finding a good linear algebra text, and learning about vector spaces and dual spaces.

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I just want to know how bra-ket notation actually works...
The terminology confuses me most times...
So... the state of the particle is written as $$\Psi$$ but as a symbol which covers all states. It's just a symbol use to represent "the state" regardless of what that state is... And $$<x_n|\Psi>$$ means the probability that $$\Psi$$ will collapse down to $$x_n$$ is $$\Lambda_n$$? I'm not sure what the mod and the exponent was for... Could you please correct me and explain that?

JDude13 said:
I just want to know how bra-ket notation actually works...
The terminology confuses me most times...
So... the state of the particle is written as $$\Psi$$ but as a symbol which covers all states. It's just a symbol use to represent "the state" regardless of what that state is... And $$<x_n|\Psi>$$ means the probability that $$\Psi$$ will collapse down to $$x_n$$ is $$\Lambda_n$$? I'm not sure what the mod and the exponent was for... Could you please correct me and explain that?

$$\Psi$$ represents the state that your particle is currently in. You can use any symbol you want, but it's the most common one you'll find in textbooks. $$\langle x_n|\Psi\rangle$$ represents the projection of that state into the position eigenstate $$x_n$$. In exactly the same way as you can take a regular 2-dimensional vector and 'project' it onto the x and y-axis to see how much of each axis it takes up, these projections tell you how much your state is in that particular eigenstate. The number obtained from one of these projections is, in general, a complex number. Quantum mechanics is set up such that the squared magnitude of this number is equal to the probability that the state is in that particular eigenstate.

JDude13 said:
And $$<x_n|\Psi>$$ means the probability...
A probability is always a real number in the interval [0,1], but the expression you wrote usually evaluates to a complex number with a non-zero imaginary part. That complex number is called a probability amplitude, or just an amplitude. You have to compute the square of its absolute value to get the corresponding probability.

What you wrote has nothing to do with bra-ket notation. It's just an inner product. In bra-ket notation, you would write $|\psi\rangle$ instead of $\psi$. Both expressions refer to the same state vector, but when you write it as $|\psi\rangle$, you call it a "ket" instead of a "state vector" or a "wavefunction". If $|\phi\rangle$ is another ket, then the corresponding bra $\langle\phi|$ is a function that takes $|\psi\rangle$ to the inner product of $|\phi\rangle$ and $|\psi\rangle$. See the post I linked to before for details.

Fredrik said:
A probability is always a real number in the interval [0,1], but the expression you wrote usually evaluates to a complex number with a non-zero imaginary part. That complex number is called a probability amplitude, or just an amplitude. You have to compute the square of its absolute value to get the corresponding probability.

What you wrote has nothing to do with bra-ket notation. It's just an inner product. In bra-ket notation, you would write $|\psi\rangle$ instead of $\psi$. Both expressions refer to the same state vector, but when you write it as $|\psi\rangle$, you call it a "ket" instead of a "state vector" or a "wavefunction". If $|\phi\rangle$ is another ket, then the corresponding bra $\langle\phi|$ is a function that takes $|\psi\rangle$ to the inner product of $|\phi\rangle$ and $|\psi\rangle$. See the post I linked to before for details.

Umm... I'm just a little too young to understand this... I know the general concepts of wave-functions but the mathematical details escape me...

Hi JDude13!

|x2> is the infinitely-long column vector (0,1,0 …)

<x2| is the infinitely-long row vector (0,1,0 …)

Ψ is an infinitely-long column vector Λ1|x1> + Λ2|x2> + Λ3|x3> + … , which is the same as (Λ123, … )

<x2|Ψ> (ie, <x2||Ψ>, but we always leave out the second | ) is the scalar (1x1 matrix) obtained by multiplying those two vectors …

obviously, it's exactly Λ2

Λ2 is a complex number, and its magnitude squared is P(x = x2)

JDude13 said:
Umm... I'm just a little too young to understand this... I know the general concepts of wave-functions but the mathematical details escape me...
In that case, you should probably focus on learning the established theories instead of developing your own. You need to study at least the basics of calculus and linear algebra if you want to understand QM. (If you really want to understand the mathematical aspects of the theory, you need to spend years studying topology and functional analysis as well).

Fredrik said:
In that case, you should probably focus on learning the established theories instead of developing your own. You need to study at least the basics of calculus and linear algebra if you want to understand QM. (If you really want to understand the mathematical aspects of the theory, you need to spend years studying topology and functional analysis as well).

I contend that the best way to learn about a topic is to question it.
And behold! I've learned about bra-ket notation and sigma notation! I'm not doing anything serious. I am very sure that it won't amount to anything as a theory. It's just keeping me occupied. It makes me very proud to look at my equations all set out on my whiteboard and to live in my own little universe governed by my own laws. XD

Anyway... How would I know which complex number is represented by $$\Lambda_n$$? And how can that equal a real probability?

JDude13 said:
Anyway... How would I know which complex number is represented by $$\Lambda_n$$? And how can that equal a real probability?

The mathematical framework for computing these sorts of probabilities is pretty much what all of Quantum Mechanics is. There's really no shortcut to understanding all of this--you have study all of the math. If you aren't interested in learning QM, at least learn some linear algebra and/or some calculus--you'll be unable to do anything with any sort of quantum theory, either conventional or homebrewed, without a solid grasp of those.

JDude13 said:
I just want to know how bra-ket notation actually works...
The terminology confuses me most times...

Try Chapter A1: ''Fundamental concepts in quantum mechanics'' of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#A1

And don't forget to read the first two sections of Chapter C4, titled: ''Learning quantum mechanics at age 14'' and ''Research at age 16''!

1. What is Bra-ket notation?

Bra-ket notation is a mathematical notation used to represent quantum states in quantum mechanics. It uses the symbols "bra" and "ket" to represent vectors and their duals, respectively.

2. How is Bra-ket notation used in layman's terms?

Bra-ket notation is used to represent the state of a quantum system, which can be thought of as a combination of different possible states. The "bra" symbol represents the state of the system in a particular basis, while the "ket" symbol represents the state of the system after a measurement has been made.

3. What is the significance of the "bra" and "ket" symbols in Bra-ket notation?

The "bra" and "ket" symbols in Bra-ket notation represent the dual nature of quantum states. The "bra" symbol represents a linear functional, while the "ket" symbol represents a vector. Together, they allow for the representation of quantum states in a concise and consistent manner.

4. How do you notate with Bra-ket notation?

To notate with Bra-ket notation, you first write the "bra" symbol, followed by the vector or state being represented. Then, a vertical bar is placed between the "bra" and "ket" symbols, and the "ket" symbol and the state being measured are written after it. For example, the notation <A|B> represents the state of the system in the A basis after measuring it in the B basis.

5. Can Bra-ket notation be used for any type of quantum system?

Yes, Bra-ket notation can be used for any type of quantum system, including qubits, multi-level systems, and infinite-dimensional systems. It is a universal notation that allows for the representation of quantum states in a clear and concise manner.

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