What is Dirac notation: Definition and 103 Discussions
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.)
If we for example have such a bipartite state:
$$  \phi > = \frac{1}{2} [ 0>0> + 1>0> + 0>1> + 1>1> ] $$
What is the easiest way to compute a density matrix of bipartite states? Should I just compute it as it is? i.e:
$$ \rho =  \phi > < \phi  $$
Or should I convert to matrix form...
a and b were fairly easy to solve; but the c part which actually demands the probability! How are we suppose to fetch the value if the function can't even be normalized; I tried to make some assumptions like making the system bounded; but I don't think that it's the right way to do so... What...
So first I rewrote H as a matrix:
$$ H =
\begin{pmatrix}
a & b \\
b & c
\end{pmatrix} $$
And tried to find the eigenvalues/energies of H, so I solved
$$ det (H  \lambda I ) =
\begin{vmatrix}
a\lambda & b \\
b & c\lambda
\end{vmatrix} = (a\lambda)(c\lambda)  b^2 = ac  a\lambda ...
Hi
If A is a linear operator but not Hermitian then the expectation value of A2 is written as < ψ  A2 ψ >. Now if i write A2 as AA then i have seen the expectation value written as < ψ  A+A ψ > but if i only apply the operators to the ket , then could i not write it as < ψ  AA  ψ > ? In...
I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in confidently interpreting his use of Dirac Notation in Section 1.9 ...
in Section 1.9 we read the following:
I need some help to confidently interpret and proceed with Neuenschwander's notation...
I am trying to convert the attached picture into dirac notation.
I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ>
The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS.
*Was going to type in LaTex but I...
Inner product is a generalization of the dot product on spaces other than Euclidean and for vectors it is defined in the same way as the dot product. If we have two vectors $v$ and $w$, than their inner product is: $$\langle vw\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n $$
where $v_1,w_1...
Shankar Prin. of QM 2nd Ed (and others) introduce the inner product:
<iV> = vi ...(Shankar 1.3.4)
They expand the ket V> as:
V> = Σ vii>
V> = Σ i><iV> ...(Shankar 1.3.5)
Why do they reverse the order of the component vi and the ket i> when they...
Starting with finding the probability of getting one of the states will make finding the other trivial, as the sum of their probabilities would be 1.
Some confusion came because I never represented the states ##\pm \textbf{z}\rangle## as a superposition of other states, but I guess you would...
hi
i was recently introduced to the Dirac notation and i guess i am following it really well , but can't get my head around the idea that the bra vector
said to live in the dual space of the ket vectors , i know about linear transformation and the structure of the vector spaces , and i realize...
hi
i am recently following the nptel course in quantum mechanics (The Course ) and it seems like a really good course , but i can't find
the book that it based on .
my question is : had anyone saw that course before to suggest a QM book related to it ?
 she began by an introduction to vector...
If we have the wavefunction ##ab \rangle##, what do the a and b stand for? In particular, do a and b signify an outcome of some pending or possible measurement, or do they signify some aspect of the wavefunction, and if so, which aspect?
Edited after post below:
Hi,
I need to show that the square of the expectation value of an observable takes a certain form in Dirac notation.
I know in wave notation that the expectation value is a sandwich integral which looks like this:
##<A>=\int_{\infty}^\infty \Psi^*(x) \hat A \Psi (x)...
Hello everyone,
I'm stuck on the question which I have provided below to do with Dirac notation:
In these questions a>, b> and c> can be taken to form an orthonormal basis set
Consider the state ξ> = α(a> − 2b> + c>). What value of α makes ξ> a normalised state?
I'm brand new to Dirac...
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...
Hi!
I am studying Shankar's "Principles of QM" and the first chapter is all about linear algebra with Dirac's notation and I have reached the section "The Characteristic Equation and the Solution to the Eigenvalue Problem" which says that starting from the eigenvalue problem and equation 1.8.3...
I am confused about the vector notation of quantum states when I have a 2 qubit system.
For 1 qubit, I just write l1> = (0 ;1 ) for representing 1,
and l0> = (1;0) for representing 0.
Dirac notation is straightforward
However when it comes to representing two qubits in linear algebra I...
Hello everyone,
I have been working through some research papers on a topic that really interests me, but I believe I am misunderstanding a few things about Dirac Notation. I have expressions that read:
\begin{align*}
&< \psi_n \mid g(H  E_{n+1}) \mid \psi_n> \text{,} \\
&< \psi_n \mid (H ...
Using that ##\hat{a} =a = \sqrt{\frac{mw}{2 \hbar}} \hat{x} +\frac{i}{\sqrt{2mw \hbar}} \hat{p}## and ## a \dagger = \sqrt{\frac{mw}{2 \hbar}} \hat{x} \frac{i}{\sqrt{2mw \hbar}} \hat{p}##
We can solve for x in term of the lowering and raising operator.
Now, recently I read a derivation of...
I want to prove Cauchy–Schwarz' inequality, in Dirac notation, ##\left<\psi\middle\psi\right> \left<\phi\middle\phi\right> \geq \left\left<\psi\middle\phi\right>\right^2##, using the Lagrange multiplier method, minimizing ##\left\left<\psi\middle\phi\right>\right^2## subject to the...
In a PDF i was looking through i came about a question
for the operator P = a><b
find Px(adjoint)
the adjoint was defined as
<vPxu> = (<uPv>)* where u and v can be any bra and ket
now for the question:
(<ua><bv>)* = <vPxu>
this is the confusing step , i thought conjugated simply...
hey guys just a quick question , within the Dirac notation I we have bras and kets.Is it allowable to simply hermitianly conjugate everything , e.g:
<wc> = <bc>  <dc>
Can we then:
<cw> = <cb> <cd>
Or is there some subtly hidden rule.
Suppose I have a 1D harmonic oscilator with angular velocity ##\omega## and eigenstates ##j>## and let the state at ##t=0## be given by ##\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates.
I've previously shown that ##\hat{H} = \sum_j...
I have the following matrix given by a basis \left1\right\rangle and \left2\right\rangle:
\begin{bmatrix}
E_0 &A \\
A & E_0
\end{bmatrix}
Eventually I found the matrix eigenvalues E_I=E_0A and E_{II}=E_0+A and eigenvectors \leftI\right\rangle = \begin{bmatrix}
\frac{1}{\sqrt{2}}\\...
I am completely baffled by bit of notation in Quantum Mechanics Concepts and Applications by Zitteli. He is trying to get the differential equation for the ground state of a harmonic oscillator using the algebraic method as opposed to Schrodinger's equation. I suspect he is compressing a lot of...
Homework Statement
The Hamiltonian of a certain twolevel system is:
$$\hat H = \epsilon (1 \rangle \langle 1   2 \rangle \langle 2  + 1 \rangle \langle 2  + 2 \rangle \langle 1 )$$
Where ##1 \rangle, 2 \rangle## is an orthonormal basis and ##\epsilon## is a number with units of...
Hello everybody,
Dirac notation uses "bras"( <a ) and "kets"( b> ), which are row vectors and column vectors respectively, but what would something like a, b> mean? It makes no notational sense to me
Context: A couple of photons going through beam splitters in an interferometer. One is...
Homework Statement
I have the following question (see below)
Homework Equations
The eigenvalue equation is Au = pu where u denotes the eigenstate and p denotes the eigenvalue
The Attempt at a Solution
I think that the eigenvalues are +1 and  1, and the states are (phi + Bphi) and (phiBphi)...
Homework Statement
I have a vector V with components v1, v2in some basis and I want to switch to a new (orthonormal) basis a,b whose components in the old basis are given. I want to find the representation of vector V in the new orthonormal basis i.e. find the components va,vb such that v⟩ =...
Two questions regarding the completeness relation:
First: I understand that the completeness relation holds for basis vectors such that ## \sum_{j=1}^{m}  n_{j} \rangle \langle n_{j}  =\mathbb{I}##. Does it also hold for unitnormalized sets of state vectors as well, where ##  \phi_{j}...
I am familiar with the derivation of the resolution of the identity proof in Dirac notation. Where ##  \psi \rangle ## can be represented as a linear combination of basis vectors ##  n \rangle ## such that:
##  \psi \rangle = \sum_{n} c_n  n \rangle = \sum_{n}  n \rangle c_n ##
Assuming an...
I promise that anytime I have question about Dirac notation I will ask it in this thread.
I do not know how to parse the following Dirac notation.
\Psi'\rangle = u\rangle U\rangle
Can someone please convert the Dirac notation to matrix notation?
Is the RHS of the conjugate relationship
Ad(g)x = gxg1
from the Lie algebra equivalent to:
<gλg>
In the Dirac notation of quantum mechanics?
I am looking at this in the context of gluons where g is a 3 x 1 basis matrix consisting of components r,g,b, g1 is a 1 x 3 matrix consisting of...
Homework Statement
Reading Sakurai I cannot see how he gets to the end of 1.7.15 as below:
Homework Equations
∫dx'x'><x'dx' α>
= ∫dx'x'>{<x' α>Δx'∂/∂x'<x' α>}
The Attempt at a Solution
I tried a Taylor expansion but cannot see how this is derived.
Homework Statement
A particle in a harmonic oscillator potential in the following state after a time t:
##  ψ(t) > = \frac{1}{\sqrt{2}} [e^{(iE_0 t/\hbar)} ψ_0> + e^{(iE_1 t/\hbar)} ψ_1> ] ##
I want to write an expression for ## <ψ(t) \hat{x}  ψ(t) > ##.
Homework Equations
The...
It's my first post so big thanks in advance :)
1. Homework Statement
So the question states "By interpreting <pxΨpxΨ> in terms of an integral over x, express <Ekin> in terms of an integral involving ∂Ψ/∂x. Confirm explicitly that your answer cannot be negative in value." ##The 'px's should...
Hello there !
I found this discussion http://physics.stackexchange.com/questions/155304/howdowenormalizeadeltafunctionpositionspacewavefunction about dirac notation and delta function .
The one that answers to the problem says that ##<aa>=1## and ##<aa>=0## .
As far as i know:
1)...
I've been working through some dirac notation and I'm stuck...
Here's where I'm at:
I understand that an expectation value: <x> = ∫ ψ* x ψ dx = <ψxψ> = <ψxψ>
Also, we can say Hψ> = Eψ> where E is an eigenvalue of the operator H and ψ> represents a state your acting on.
I get that you can...
I'm just learning about the whole Dirac notation stuff and I have come across the fact that bra's and ket's are somehow independent of bases. Or rather that they do not need the specification of a basis. I really don't understand this from a vector point of view. Maybe that is the problem...
Homework Statement
Rewrite the state ψ⟩ = √(1/2)(0> + 1>) in the new basis.
3⟩ = √(1/3)0⟩ + √(2/3)1⟩
4⟩ = √(2/3)0⟩ − √(1/3)1⟩
You may assume that 0⟩ and 1⟩ are orthonormal.
Homework Equations
The Attempt at a Solution
[/B]
I have a similar example in my notes however there...
Homework Statement
Consider a onedimensional particle subject to the Hamiltonian H with wavefunction \Psi(r,t) =\sum_{n=1}^{2} a_{n}\Psi _{n}(x)e^{\frac{iE_{n}t}{\hbar}}
where H\Psi _{n}(x)=E_{n}\Psi _{n}(x) and where a_{1} = a_{2} = \frac{1}{\sqrt{2}}. Calculate the expectation value of the...
I'm looking through my lecture notes, (studying relativistic corrections/perturbation theory using hydrogen), and I seem to have a mind block with one of the equations (the last one from the 3 in the middle).
I know that the kinetic energy and coulomb potential has been subbed in for the...
Hi guys,
I m reading some theoretical physics paper that requires knowledge of dirac notation if someone could point me out to a good tutorial on it I come from a math background but I am studying this paper with my supervisor.
Homework Statement
This isn't really a problem so much as me not being able to see how a proof has proceeded. I've only just today learned about Dirac notation so I'm not too good at actually working with it. The proof in the book is:
Z> = V>  <WV>/W^2W>
<ZZ> = <V  ( <WV>/W^2 ) W...
Quantum Mechanics using Index notation. Is it possible to do it?
I really don't get the Dirac Notation, and everytime I encounter it, I either avoid the subject, or consult someone who can read it. There doesn't seem to be any worthy explanation about it, and whenever I ask what is the Hilbert...
Homework Statement
This makes intuitive sense to me, but I am getting stuck when trying to read the Dirac notation proof.
Anyway, the author (Shankar) is just demonstrating that the product of two operators is equal to the product of the matrices representing the factors.
Homework Equations...
Question and symbols:
Consider a stateε> that is in a quantum superposition of two particleinabox energy eigenstates corresponding to n=2,3, i.e.: ε> = ,[1/(2^.5)][2> + 3>], or equivalently:
ε(x) = [1/(2^.5)][ψ2(x) + ψ3. Compute the expectation value of momentum: <p> = <ε\widehat{}pε>...
Homework Statement
Using Dirac Notation prove for the Hermitian operator B acting on a state vector ψ>, which represents a bound particle in a 1d potential well  that the expectation value is <C^2> = <CψCψ>.
Include each step in your reasoning. Finally use the result to show the...
QUESTION
A quantum mechanical system has a complete orthonormal set of energy eigenfunctions,
n> with associate eigenvalues, En. The operator \widehat{A} corresponds to an observable such that
Aˆ1> = 2>
Aˆ2> = 1>
Aˆn> = 0>, n ≥ 3
where 0> is the null ket. Find a complete...