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

[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.

 

[tiny-tim]

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(x_n = Λ_n))

 

[JDude13]

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?

 

[tiny-tim]

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 …

x_n is, of course, position.

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

 

[JDude13]

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.

 

[tiny-tim]

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

 

[JDude13]

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.

 

[tiny-tim]

Right?

Right.

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

We do it all the time!

 

[Fredrik]

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).

This is my standard answer about bras and kets.

 

[tiny-tim]

Hi Fredrik!

I just don't know how you could write that …

… i.e. it's just a vector written in a funny way. …

… without a smilie!

 

[Fredrik]

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

 

[Chopin]

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.

 

[JDude13]

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?

 

[Chopin]

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.

 

[Fredrik]

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.

 

[JDude13]

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...

 

[tiny-tim]

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 (Λ₁,Λ₂,Λ₃, … )

<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)

 

[Fredrik]

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).

 

[JDude13]

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?

 

[Chopin]

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.

 

[A. Neumaier]

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''!