Bra–ket notation Puns

Hello. In bra-ket notation, <x|x> is an inner product, |x><x| is an outer product, and |x>|x> is a tensor product. What is <x|<x| ?

QFT courses build up complex practical notation, so that they can focus on illustrating the elegance and symmetry of QFTs, and extract useful results from them.

I’m looking for the opposite, stupid uninterpretable brute force. I’ll explain why at the end.

I want to expand out the Schrödinger equation for some QFT, say QED or the SM:

dψ/dt= (1/ iℏ) H ψ

Can I write ψ as an (infinite) vector of variables? Can I write H as an (infinite) matrix of constants? Can I multiply these out to give an infinite list of coupled first-order differential equations?

Notice how ‘yes you can do that but it’s stupid’, and ‘no you cannot do that, those aren’t even vaguely the right mathematical objects’ are profoundly different answers.

I don't understand certain 'projection' aspects of the bra-ket notation. There's this problem where we are given a wavefunction of a particle in an infinite well of some shape (not necessarily square)

psi(x) = c_1 cos(w_1 x) + c_2 sin( w_2 x) + c_3cos(w_3 x)

I know the value of the 'c's and the 'w's. We are required to calculate the possible measurable momentums and with which probabilities.

The way it was explained, since the eigenstates of the Hamiltonian (energy) can't be the same for the momentum, then momentum must have a continuous distribution (o..kay, thanks professor). The probability to get some momentum is PdpP which can be calculated through |<p|psi(x)>|^2 (i THINK i understand this) and that there are respective associated equations <p|x> = e^{ipx/hbar} and <x|psi(x)> = psi(x)

so the idea would be (for a psi(x) that becomes 0 outside (-a,a) )

<p|psi(x)> = <p|x><x|psi(x)> = <x|p>* <x|psi(x)> = int_{-a}^a

At hat point, bracket notation lost me. Any help is appreciated, thank you!

A wave-function can be defined using bra-ket notation as follows: ψ(r) = <r|ψ>

|r&gt; will take a different value at each position so let's consider a single point in space: **r_**0. We now have ψ0 = ψ(**r_**0) = <r_0|ψ>.

ψ0 is just a scalar.

What is one possible vector representation |r&gt; ? I realise there are many. How many dimensions does |r&gt; have? Could I represent this as (x0, y0, z0) or must |r_0> be some infinite dimensional things with infinitesimal numbers at each index?

Back to |r&gt;, is this a vector where each value is a function r_n(x,y,z) ?

I know how to take a conjugate transpose of a vector so the next part is easy.

Alternatively, should I avoid thinking of kets as infinite dimensional vectors with numbers for entries?

Hi all, i am an engineer trying to get to grips with the elementary concepts of QM.

I have been assigned an homework during the lecture, and I am struggling with the following question:

Question

This is my trial of answering it:

Answer

I am quite sure about the idea of using Pauli's decomposition and the properties of the Pauli matrices, while I have some doubts regarding the decomposition of the composite system density matrix...therefore I kindly ask you to help me understand where I may have done a mistake! (I guess I made one since I have no idea on how to show that the final quantity I obtained is non-negative, as the only property I have is that a, b's norms are <= 1)

Thanks very much for your help.

P.S. the various explanations are quite poorly written, sorry for that!

My quantum mechanics prof is teaching bra-ket notation, but only wants us o get our feet wet; essentially, be able to use it, but nothing more. No hilbert space. I'm having difficulty finding sources of practice on using bra-ket notation and i was hoping that someone could point me to a good source

Given that it represents an inner product, why isn't it just written as such? For example, what's the advantage of using <φ|ψ> instead of <φ,ψ>, <φ|A|ψ> instead of <φ,Aψ>, etc.?

I've been in a couple QM classes, and usually the instructor gives a brief (5-10min) introduction of the notation, and then dives right into the physics.

I know bits and pieces, but I feel like often times I get bogged down in the notation. For this reason, I'm looking for a resource (book or web) that would help me build intuition for the notation.

FWIW, I'm a first year graduation student, enrolled in my first graduate QM course, so I definitely can't half-ass it this time around.

Thanks!

In high school (IB, if anyone's familiar) we doing an exploration in maths, where I can learn about a topic of my chosing. I thought it would be really cool to learn some maths used in quantum physics. I was wondering, from those who already know it, if Bra ket (or Dirac (?)) notation is an interesting and relevant thing to learn? Would you have other recommendations instead? Any resources for learning the notation? Thank you!

http://www.amazon.ca/Lectures-Quantum-Mechanics-Steven-Weinberg/dp/1107028728

One of my professors said that we will have to recreate a proof on an upcoming exam involving all listed in the title. all he really told us is that we will need to know that <raising operator> + <lowering operator> is proportional to 2x while <raising operator> - <lowering operator> is proportional to 2ip. (with x and p being vector quantities)

I tried to find if something like this online but ultimately failed. I was hoping maybe something like this rings a bell to someone else?

In quantum mechanics, the teacher told us to prove that the momentum operator is hermitian. I used operators and bra-ket notation during last year, its not anything new to me, but I was curious about if there's any way to prove it just with this notation and I couldn't find any, just the typical one:

https://www.quora.com/How-do-you-prove-the-momentum-operator-is-Hermitian

Let's say we have 2 level system with its Hamiltonian operator H and two eigenvectors with [;|\phi_1>;] and [;|\phi_2>;] eigen vectors of the system. Their eigen values respectively will be E1 and E2.

One of the elements of the matrix (specifically row 2 column 1) that represents H in using the eigenvectors as basis vectors is then :

[;<\phi_2|\hat{H}|\phi_1>;]

[;<\phi_2|\cdot(\hat{H}|\phi_1>)=E_1<\phi_2|\phi_1>=0;] since the two vectors are orthogonal. So far so good.

and then due to the associative property I can just as well say:

[;(<\phi_2|\hat{H})\cdot|\phi_1>=E_2*<\phi_2|\phi_1>=0;] but [;E_2;] is an eigen value so it has to be real meaning:

[;(<\phi_2|\hat{H})\cdot|\phi_1>=E_2<\phi_2|\phi_1>=0;]

and since they're two different ways of saying the same thing we have :

[;E_2<\phi_2|\phi_1>=E_1<\phi_2|\phi_1>;]

So at first I was tempted to think that this would imply [;E_1=E_2;] but this is a mathematical fallacy since I can't divide through by 0 which is fine.

Is everything I did regarding the BraKet notation kosher though?

edit: typo

The Schrodinger equation can be written as

iħ*d/dt(|Ψ>)=Ĥ|Ψ>.

My QM book has a line which implies that

-iħ*d/dt(<Ψ|)=<Ψ|Ĥ

is also true.

This isn't obvious to me and I'm trying to work it out. So far I think I've determined that it is true as long as you can somehow treat the derivative as an operator and that operator is self-adjoint. This is where I get confused and lost since I'm not sure how to go about determining that or if what I'm saying even makes sense.

I'm honestly just not very comfortable with the notation yet so I'm probably missing something very obvious.

So, I've been reading up on some Wikipedia articles on bra-ket notation, spin, etc. And I noticed that the difference between bra-ket notation and normal vectors is that bras <A| are complex conjugates with kets |B>. I can understand a usefulness of complex conjugates with respect to pure mathematics, but what are some examples of variables in physics that use complex or imaginary values?

I am going through Shankar's Introduction to Quantum Mechanics. It's a great book but I seem to have missed some of the stuff with bra-ket notation early on and now most of the stuff we're currently learning in class is written in it. I'm okay with going back and rereading the math section but I was hoping there might be a quicker alternative available for someone who just needs some quick catch-up. What do you think?

Anyone recognise the BRA-KET Notation in math as related to TOP? The notation is the Empty Set (zero with the line through it) followed by the PSI (SAI) symbol. There seems to be so much related to math, numbers, design, energy, realities, science, etc, in Dema.

Noob question. So I know a ket is a vector. What does |0> and |1> mean? Is |0> just a vector of 0’s and |1> just a vector of 1’s?

Many thanks in advance.

Note: I can’t type out the ket notation properly on my phone, so the ket notation used here might look a little funny.

I'm trying to write the following expression, and it looks ugly.

\begin{equation}\label{eq:2}
	<i|\hat{A}|j> = \int dr_i d\omega_i\Psi_{i/2}^*(r_i)\sigma_i^*(\omega_i)\hat{A}\Psi_{j/2}(r_i)\sigma_j(\omega_i)
\end{equation}
\begin{equation}\label{eq:3}
	<k|\hat{B}|l> = \int dr_k d\omega_k\Psi_{k/2}^*(r_k)\sigma_k^*(\omega_k)\hat{B}\Psi_{l/2}(r_k)\sigma_l(\omega_k)
\end{equation}

How it compiles:

https://imgur.com/FovgVd2

Anyone got an idea what to do so that things are better aligned, and spacing is less wack?

Is it just a flip of where to put the asterick (implying complex conjugate) in equations?

Hello all,

I'm learning quantum computation, and I'm finding bra-ket notation a little confusing.

Suppose there's a wavefunction named ψ. I usually see this written as |ψ⟩ to indicate, roughly, that it's a wavefunction / vector in the Hilbert space. But if I have a particle that's in the n'th state, where n is an integer, this is also written as |n⟩. So in the first case, the symbol in the ket is a name, and in the second case it's an integer used as an index. In the context of quantum computation this is especially weird because you have a lot of classical states that get promoted to wavefunctions as eg |F(x)+G(y)⟩, where x and y are variables in a classical computer, and F and G are classical functions. It also seems cumbersome to write the difference of wavefunctions as ||ψ⟩-|ψ'⟩| instead of |ψ-ψ'|.

Is there a convention for making this less verbose or ambiguous? Like always writing column-vector ψ without the ket, and leaving |foo⟩ = foo'th element of the computational basis? Or would that be confused for a scalar named ψ?

Thanks!

Hi I have a Quantum Mechanics exam soon and whenever I see bra ket notation with Schrodinger equations my mind just falls apart. Does anyone maybe have some old notes or materials on how to approach them? It would save my life

Firstly I am quite new to QM. Lets say |A> is a quantum state and it is |A> = alpha1 * |0> + alpha2 * |1> So I understand check whether a state |A> is normalised, you calculate whether <A| A> is equal to 1 and if it is not, you divide |A> by sqrt(<A|A>) to find the normalised version of |A>.

However after the density operator phi is introduced, I was taught that from now on, phi will be used to represent a quantum state and lets say PHI = alpha1 * |0> <0| + alpha2 * |0><1| + alpha3 *|1><0| + alpha4 * |1><1|. But based on my calculations sqrt(<PHI|PHI>) is not a scalar. So how do I determine whether it is normalised or not?

Noob question. So I know a ket is a vector. What does |0> and |1> mean? Is |0> just a vector of 0’s and |1> just a vector of 1’s?

Many thanks in advance.

Note: I can’t type out the ket notation properly on my phone, so the ket notation used here might look a little funny.

Noob question. So I know a ket is a vector. What does |0> and |1> mean? Is |0> just a vector of 0’s and |1> just a vector of 1’s?

Many thanks in advance.

Note: I can’t type out the ket notation properly on my phone, so the ket notation used here might look a little funny.