Bra-ket Definition and 62 Threads

I suspect it will help if you know about my background: I did some linear algebra in university but never used it and am now in my mid 60s. I am interested in understanding the mathematics of quantum physics. I have read a number of layman's texts on quantum mechanics, but they all gloss over...

What's the difference between a bra vector and ket vector in specifying spin states except for notational convenience when calculating probablility amplitudes? Are they equivalent?

It's said that every ket has a unique bra. For any vector ##|v> ∈ V## there is a unique bra ##<v| ∈ V*##. I'm not sure what that means. What is unique? Can anyone please help me understand. Thank you

Firstly, apologies for the latex as the preview option is not working for me. I will fix mistakes after posting. So for ##<x>## = (##\sqrt{\frac{\hbar}{2m\omega}}##) ##(< \alpha | a_{+} + a_{-}| \alpha >)## = (##\sqrt{\frac{\hbar}{2m\omega}}##) ##< a_{-} \alpha | \alpha> + <\alpha | a_{-}...

It's easy to show that ##[\Delta A, \Delta B] = [A,B]##. I'm specifically having issues with evaluating the bra-ket on the RHS of the uncertainty relation: ##\langle \alpha |[A,B]|\alpha\rangle = \langle \alpha |\Delta A \Delta B - \Delta B \Delta A|\alpha\rangle## The answer is supposed to be...

I am trying to solve for the uncertainty in energy ##\Delta E## in the following exercise: $$\Delta E = \sqrt{\langle \Phi | (\hat H - \bar E )^2 | \Phi \rangle}$$ Questions What does ##(\hat H - \bar E )^2## mean? Is it a simple binomial expansion into ##\hat H^2 - 2 \bar E \hat H + \bar...

Could somebody show me how to derive this equation? How can I get right side from left. Step by step, thanks...

Homework Statement I am stuck on the second paragraph but I thought I would add the first paragraph in case some context would help! Homework Equations |A> = cos(theta)|H> + sin(theta)|V> The Attempt at a Solution I am not wholly comfortable with bra-ket notation with the outer product p =...

Say I have a vector product |x+a⟩⟨x| and I multiplied it by a ket vector |x'⟩. Can I pull the |x'⟩ into the ket vector |x+a⟩? also could you split up the ket vector |x+a⟩ into two ket vectors added together?

In my lecture notes, it says that ##\left\langle l \right| A_{nm} \left| \psi \right\rangle = \sum_{n,m} A_{nm} \left\langle m \right| \left|\psi \right\rangle \left\langle l \right| \left| n \right\rangle## ##=\sum_{n,m} A_{nm}\left\langle m \right| \left| \psi \right\rangle \delta_{ln}## ##=...

Question Consider the matrix $$ \left[ \matrix { 0&0&-1+i \\ 0&3&0 \\ -1-i&0&0 } \right] $$ (a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by |a1>, |a2>, |a3>. Any degenerate eigenvalues? (b) Show that the eigenvectors |a1>, |a2>, |a3> form an...

Hi! First of all I want apologize for my bad english! Second, I'm doing a physical chemystry course about the main concepts of quantum mechanics ! The Professor has given to me this definition of "the adjoint operator": <φ|Aψ> = <A†φ|ψ> My purpose is to verificate this equivalence so i...

I'm new to bra-ket notation and am slightly confused; given an infinite square well with eigenvectors: \phi = \sqrt{2/a}\sin( (n\pi x)/a) And we assume the form: H |φ> = E_n |φ> How would you then represent φ in terms of a column matrix, because that what I thought |φ> represents. Given...

Homework Statement A particle is in the state |\psi \rangle = \frac{1}{{\sqrt 3 }}|U\rangle + \frac{{a\sqrt {(2)} }}{{\sqrt {(3)} }}i|D\rangle . The up state |U\rangle = \left( {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right) and the down state |D\rangle = \left(...

Homework Statement Homework EquationsThe Attempt at a Solution Hello, I just want to make sure I am doing this right $$<a|b> = a_{x}^{*}b_{x} + a_{y}^{*}b_{y} + a_{z}^{*}b_{z}$$ $$= [(1-i)|x>][-i|x>] + (2 |y>)(-3 |y>) + (0|z>)(|z>)$$ $$=(-i + i^{2})|x> - 6 |y> + 0|z>$$ $$=(-1-i)|x> - 6 |y>...

I have been studying quantum physics for some time, yet I still cannot seem to understand the principles behind bra-ket notation (especially how spin states are described and the probabilities/eigenvalues). It would be great if someone could give me a basic explanation and/or maybe recommend...

I'm new to the concepts of quanum mechanics and the bra-ket representation in general. I've seen in the textbook that the compleatness relation is used all the time when working with the bra and kets. I'm a bit confused about how this relation is being used when applied more than once in a...

I have a question on the algebra involved in bra-ket notation, eigenvalues of \hat{J}_{z}, \hat{J}^{2} and the ladder operators \hat{J}_{\pm} The question has asked me to neglect terms from O(ε^{4}) I am using the following eigenvalue, eigenfunction results, where ljm\rangle is a...

Homework Statement Show that the dot product in two-dimensional space is linear: <u|(|v> + |w>) = <u|v> + <u|w> The Attempt at a Solution I feel like I'm missing some grasp of the concept here ... I would think to just distribute the <u| and be done in that one step, but I'm being...

Homework Statement On Wikipedia there is an article about perturbation theory. To understand something I need to understand the following relation. They say: Homework Equations H |n> = E_n |n> So: <n| H = <n| E_n H is Hermitian. So: Why is this? The Attempt at a Solution...

http://i.imgur.com/ORtBJdT.jpg i don't understand why the old base is written in terms/as a linear combination of the new bases. wouldn't i want to map my coordinates from old to new not new to old?.. here's what my textbook says about it, can you guys interpret this for me, i still don't...

forgive the messiness; i take bad notes in class. http://i.imgur.com/VmW8Ubg.jpg towards the middle of the page where it says "this is equivalent to..." and then my professor wrote what follows but i thought the row vector should be complex conjugates? ie, the red writings are not actually...

so I'm fine with the kets, e.g, |a>.. but i don't understand what the bras are. the kets are basically just a column vector right? ie the components (with the direction) of the vector being described. but what is the bra? this was given to us in class: <a|=a1<e1|+a2<e2|= (a1* a2*) (where e1...

Hello, I am working my way though Sakurai's book on Quantum MEchanics and am having some problems understanding the bra-ket notation. I keep believing I understand everything there is to it but then he will do something in a single line that I cannot understand. This is one of them. If...

Homework Statement Confirm explicitly that ##\frac{1}{2m}\langle \hat{p}_x \Psi | \hat{p}_x \Psi \rangle## cannot be negative. Homework Equations ##-i\hbar \frac{\partial}{\partial x} = \hat{p}_x## The Attempt at a Solution i seem to get: ##\frac{1}{2m}\langle \hat{p}_x \Psi | \hat{p}_x...

I am having trouble understanding the following: Uf: |x>|y> → |x>|y \oplusf(x)> \oplus being a mod 2 operation (nand)? I suppose I don't understand how to read the "ket" states so well. As far as I understand we have that since x and y can be 0,1 only if |x=1>|y=1> then if f(x) = 1 then...

Homework Statement sorry about the lack of LaTex but I don't know how to do bra-ket notation in tex vectors |1> and |2> are a complete set of normalized basis vectors. the hamiltonian is defined as |1><1|-|2><2|+|1><2|+|2><1| find the eigenvalues and eigenvectors in ters of |1> and |2>Homework...

I'm continuing through Dirac's book, The Principles of Quantum Mechanics. You can view this as a google book in the link below. http://books.google.com.au/books?id=...page&q&f=false On page 28-29 he proves this Theorem: If ξ is a real linear operator and ξm|P> = 0 (1) for a...

Homework Statement Consider a three-dimensional vector space spanned by an orthonormal basis |1\rangle, |2 \rangle, |3 \rangle . Kets |\alpha \rangle, |\beta \rangle are given by |\alpha \rangle = i|1\rangle -2|2 \rangle -i|3\rangle, \qquad |\beta \rangle = i|1\rangle +2 |3\rangle. part...

What are the rules of analytical – not numerical (matrix) entry of bra-ket convertion – operations on bra-ket, in particular – tensor product ? For example – how in analytical form to do this: U|\Psi\rangle where: U=I\otimesI I=|0\rangle\langle0|+|1\rangle\langle1|...

Sorry for disregarding the template; I'm not really working out a homework problem as much as just trying to follow the reasoning in the text. I'm working through the first chapter of Quantum Mechanics, McIntyre, and I'm a little bit confused by the following. The text introduces bra-ket...

bra - ket?? Hi, maybe a stupid question, but i would like to know if, if We have a real number, but we are i a vector space, and the operator is hermitian, is |a> is equal to < a |*? i assume this, because if a is the vector (1,0) (spin up), and only real entries. im trying to make...

Hi If C is an operator such that C|1> = |1> and C|2>=|2>, then C^3 |1>= |1>|1>|1> =|1> ^ 3 ? If yes, then what does this C^3 represent? :confused:

I'm a little frustrated with the quantum m lectures I've been watching. I've watched Susskind's, one in India and now James Binney's, as well as read about 3 books. They all teach this Bra-ket notation and in none of the three books I have on worked problems do they every give you a chance to...

I am confused about the the notation |ab> for an entangled pair. Isn't this the same as the tensor product |a> \otimes |b>? If so, I run into another confusion when using the corresponding matrices. I read that I should apply a Hadamard operator H twice to the input state |01>. Does this mean...

Hello all, Homework Statement I’m trying to derive a result from a book on quantum mechanics but I have trouble with bra-ket notation and operators… Let’s say we have a photon moving along the cartesian z-axis. It is polarized and its state is Psi(theta) = cos (theta) x1 + sin(theta) x1...

Hi guys, I'm having some trouble with bra-ket algebra. For example, our lecturer did on the board, <Sx+|Sz|Sx+> So what I would do is, ignoring any factors of 1/sqrt(2) or 1/2 or hbar. Sx+ = |+> + |-> Sz = |+><+|-|-><-| => ( |+> + |-> )(|+><+|-|-><-|)( |+> + |->) This is...

Homework Statement The question is to evaluate the expression e^-iA, where A is a Hermitian operator whose eigenvalues are known (but not given) using bra-ket algebra. Homework Equations See above. The Attempt at a Solution I have been looking around, reading the textbook and...

What am I doing wrong here? Let \psi be a ket whose representation in the X basis is given by \psi(x)\ =\ \langle x|\psi\rangle\ =\ e^{-x^{2}/2} Then \psi(-x)\ =\ \langle -x|\psi\rangle\ =\ e^{-x^{2}/2}\ = \psi(x) (1) But we also have: \psi(-x)\ =\ \langle -x|\psi\rangle (2) \ =\...

|t> + |a> = ?? As an angle from the transition axis now I know it is 45 degrees is the answer but I am not sure what |t> or |a> equals. I know |theta> = cos theta |t> + sin theta |a> so how do I go from here? Does |t> = cos^2 theta and |a> = sin^2 theta? Thanks. Stephen

This is a pretty trivial question, but how is the Schrodinger equation written out in full, time dependency and all in Dirac notation? I'm interested in this from a purely aesthetic point of view but I'm also a bit confused as to what the bras and the kets really are.

Hi, i am evaluating the integral \int_{-\infty}^{+\infty}dE \langle p'|E \rangle \langle E| e^{-iEt/ \hbar} |p\rangle However, i am unsure how to evaluate \langle E| e^{-iEt/ \hbar} |p\rangle . I am not sure if it is simply e^{-iEt/ \hbar} \times \langle E|p\rangle or something else. Any...

Homework Statement <e^{ip'x}|x^{2}|e^{ipx}> Homework Equations The Attempt at a Solution Its pretty obvious that its difficult to integrate in position-space, so I rewrite x in momentum space (i.e. the second-order differential operator with respect to p). If that is...

Hello, i am a beginner in quantum mechanics and i have those basic questions on the bra-ket notation: Which dimension has a ket | \phi > describing a state normally? Maybe \quad C ^n? Which dimension has a bra-ket <\psi | \phi >then? Maybe \quad C ? How do you get the matrix...

Hello everybody, why is the adjoint of a bra-ket like this: < \phi | \psi >^+ = < \psi | \phi > Is it a definition or can it be derived somehow? Thanks :)

We know that < \phi | \psi >* = < \psi | \phi > where * denotes the complex conj. so if \psi and \phi are ordinary real valued functions (as opposed to matrices or complex valued whatevers) can we also say: < \phi | \psi > = < 1 |\phi \psi > = <\phi \psi | 1> Or what if \phi = \psi...

I am new to qm and very new to bra-ket notation. If you, as a physicist, saw this: |\phi>=\Sigma(\sqrt{\Lambda_n}|x=x_n>) what would you understand about the system it is describing?

Homework Statement I've solved my problem now. I was trying to show that LHS=RHS: (|+><-| + |-><+|)^2 = (|+><+| + |-><-|) this can be done by using <-|->=1 (normalization) and <x|->=0 (orthogonal). LHS: (|+><-||+><-|) + (|+><-||-><+|) + (|-><+||+><-|) + (|-><+||-><+|) = 0 + |+><+| + |-><-|...

Hey physicists! I'm having trouble getting my head around bra-ket algebra and was wondering if anyone knows any problems/worked solutions to help me understand it. Either that or some tutorial resources. Thanks, Andrew

Question is in attachment The Attempt at a Solution i) < 5 | 3 > ii) G < 3 | 3 > iii) (Lz)/2 < 5 | 5 >