In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "
⟨
{\displaystyle \langle }
" and "
⟩
{\displaystyle \rangle }
", and a vertical bar "

{\displaystyle }
", to construct "bras" and "kets" .
A ket looks like "

v
⟩
{\displaystyle v\rangle }
". Mathematically it denotes a vector,
v
{\displaystyle {\boldsymbol {v}}}
, in an abstract (complex) vector space
V
{\displaystyle V}
, and physically it represents a state of some quantum system.
A bra looks like "
⟨
f

{\displaystyle \langle f}
", and mathematically it denotes a linear form
f
:
V
→
C
{\displaystyle f:V\to \mathbb {C} }
, i.e. a linear map that maps each vector in
V
{\displaystyle V}
to a number in the complex plane
C
{\displaystyle \mathbb {C} }
. Letting the linear functional
⟨
f

{\displaystyle \langle f}
act on a vector

v
⟩
{\displaystyle v\rangle }
is written as
⟨
f

v
⟩
∈
C
{\displaystyle \langle fv\rangle \in \mathbb {C} }
.
Assume on
V
{\displaystyle V}
exists an inner product
(
⋅
,
⋅
)
{\displaystyle (\cdot ,\cdot )}
with antilinear first argument, which makes
V
{\displaystyle V}
a Hilbert space. Then with this inner product each vector
ϕ
≡

ϕ
⟩
{\displaystyle {\boldsymbol {\phi }}\equiv \phi \rangle }
can be identified with a corresponding linear form, by placing the vector in the antilinear first slot of the inner product:
(
ϕ
,
⋅
)
≡
⟨
ϕ

{\displaystyle ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi }
. The correspondence between these notations is then
(
ϕ
,
ψ
)
≡
⟨
ϕ

ψ
⟩
{\displaystyle ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\equiv \langle \phi \psi \rangle }
. The linear form
⟨
ϕ

{\displaystyle \langle \phi }
is a covector to

ϕ
⟩
{\displaystyle \phi \rangle }
, and the set of all covectors form a subspace of the dual vector space
V
∨
{\displaystyle V^{\vee }}
, to the initial vector space
V
{\displaystyle V}
. The purpose of this linear form
⟨
ϕ

{\displaystyle \langle \phi }
can now be understood in terms of making projections on the state
ϕ
{\displaystyle {\boldsymbol {\phi }}}
, to find how linearly dependent two states are, etc.
For the vector space
C
n
{\displaystyle \mathbb {C} ^{n}}
, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and operators are interpreted using matrix multiplication. If
C
n
{\displaystyle \mathbb {C} ^{n}}
has the standard hermitian inner product
(
v
,
w
)
=
v
†
w
{\displaystyle ({\boldsymbol {v}},{\boldsymbol {w}})=v^{\dagger }w}
, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the Hermitian conjugate (denoted
†
{\displaystyle \dagger }
).
It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator
σ
^
z
{\displaystyle {\hat {\sigma }}_{z}}
on a two dimensional space
Δ
{\displaystyle \Delta }
of spinors, has eigenvalues
±
{\displaystyle \pm }
½ with eigenspinors
ψ
+
,
ψ
−
∈
Δ
{\displaystyle {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{}\in \Delta }
. In braket notation one typically denotes this as
ψ
+
=

+
⟩
{\displaystyle {\boldsymbol {\psi }}_{+}=+\rangle }
, and
ψ
−
=

−
⟩
{\displaystyle {\boldsymbol {\psi }}_{}=\rangle }
. Just as above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular when also identified with row and column vectors, kets and bras with the same label are identified with Hermitian conjugate column and row vectors.
Bra–ket notation was effectively established in 1939 by Paul Dirac and is thus also known as the Dirac notation. (Still, the braket notation has a precursor in Hermann Grassmann's use of the notation
[
ϕ
∣
ψ
]
{\displaystyle [\phi {\mid }\psi ]}
for his inner products nearly 100 years earlier.)
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 braket 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...
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 braket 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 $$
\left[
\matrix
{
0&0&1+i \\
0&3&0 \\
1i&0&0
}
\right]
$$
(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":
<φAψ> = <A†φψ>
My purpose is to verificate this equivalence so i...
I'm new to braket 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)} }}iD\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
$$<ab> = a_{x}^{*}b_{x} + a_{y}^{*}b_{y} + a_{z}^{*}b_{z}$$
$$= [(1i)x>][ix>] + (2 y>)(3 y>) + (0z>)(z>)$$
$$=(i + i^{2})x>  6 y> + 0z>$$
$$=(1i)x>  6 y>...
I have been studying quantum physics for some time, yet I still cannot seem to understand the principles behind braket 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 braket 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 braket 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 twodimensional space is linear:
<u(v> + w>) = <uv> + <uw>
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 braket 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...
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 braket notation in tex
vectors 1> and 2> are a complete set of normalized basis vectors.
the hamiltonian is defined as 1><12><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 2829 he proves this Theorem:
If ξ is a real linear operator and
ξmP> = 0 (1)
for a...
What are the rules of analytical – not numerical (matrix) entry of braket convertion – operations on braket, 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 braket...
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...
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 Braket 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 braket notation and operators…
Let’s say we have a photon moving along the cartesian zaxis. It is polarized and its state is
Psi(theta) = cos (theta) x1 + sin(theta) x1...
Hi guys, I'm having some trouble with braket algebra.
For example, our lecturer did on the board, <Sx+SzSx+>
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 braket 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 Ep\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 positionspace, so I rewrite x in momentum space (i.e. the secondorder 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 braket notation:
Which dimension has a ket  \phi > describing a state normally? Maybe \quad C ^n?
Which dimension has a braket <\psi  \phi >then? Maybe \quad C ?
How do you get the matrix...
Hello everybody,
why is the adjoint of a braket 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 braket 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 braket algebra and was wondering if anyone knows any problems/worked solutions to help me understand it. Either that or some tutorial resources.
Thanks,
Andrew