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.)
The wavefunction of ##\psi\rangle## is given by the bra ket
##\psi (x,y,z)=
\langle r \psi\rangle##
I can convert the wavefunction from Cartesian to polar and have the wavefunction as ## \psi (r,\theta,\phi)##
What bra should act on the ket ##\psi\rangle## to give me the wavefunction as ##...
"##
\left[\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}+V(x)\right]\langle x \mid E\rangle=E\langle x \mid E\rangle##
is often referred to as the timeindependent Schrödinger equation in position space. This equation also results from projecting the energy eigenvalue equation...
I can't follow how the above argument leads to (1.8).
I am able to prove it only if I can show ##\langle a \mid c\rangle\langle b+c\rangle=(\langle a+\langle b) c\rangle## but I don't understand why the bra transformations <P ,<Q obey
(<P+ <Q)x = <Px + <Qx .
Is it an assumption...
We denote a scalar product of two vectors ##a, b## in Hilbert space ##H## as $(a,b)$.
In Bra Ket notation, we denote a vector a in Hilbert space as ##a\rangle##. Also we say that bras belong to the dual space ##H##∗ .
So Bras are linear transformations that map kets to a number.
Then it...
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...
Homework Statement
Let ##\vec{e}\in\mathbb{R}^3## be any unit vector. A spin ##1/2## particle is in state ##\chi \rangle## for which
$$\langle\vec{\sigma}\rangle =\vec{e},$$
where ##\vec{\sigma}## are the PauliMatrices. Find the state ##\chi\rangle##
Homework Equations :[/B] are all given...
Homework Statement
If I had two vectors say ⟨emf⟩⟨fem⟩ does this equal ⟨emf⟩2? e is a basis and f is some arbitrary function. I ask this because I have a problem which is to show the following: Show that for the Fourier expansion of f⟩ in terms of Fourier basis vectors em⟩ is...
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...
Dear All,
I am trying to understand what operators actually mean when deriving the definition of green's function. Is this integral representation of an operator in the ##xbasis## correct ?
## D = <x\int dxDx>##
I am asking this because the identity operator for nondenumerable or...
Just checking (while trying to prove the Schwarz inequality for $<fHg>$, I know $ <fg>=<gf>^* $ please confirm/correct :
If $ \psi=f+\lambda g, \:then\: \psi^*=f^*+\lambda^* g^* $
Is $ <f^*g>=<g^*f>^* $ and $ <f^*Hg>=<g^*Hf>^* $ (H hermitian)?
Is $ <f^*Hg><g^*Hf> = ...
I am trying to understand the mathematics of quantum eraser experiments, in order to deepen my understanding of what is really happening. The paper I am currently working on is:
"A doubleslit quantum eraser" by S. P. Walborn, M. O. Terra Cunha, S. Padua, and C. H. Monken (2001)
in which a...
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...
Hello everyone, I have thi doubt:
If I have a state, say psi1, associated with the energy eigenvalue E1, the integral over a certain region gives me the probability of finding the particle in that region with the specified energy E1. Now if I put an operator between the states I obtain its mean...
I'm trying to apply BRA KET notation to my notes on particle physics.
please could someone confirm that the kroneker delta function may be written
\delta _{ij} = \left \langle i j \right \rangle
OR would it be written
δij = i> <j
I know i and j are indices, so can BRA KET even be...
In class we went through the derivation of the energy of a perturbed system, I've dug my old notes out and found a bra ket derivation of the same thing, there's just one step that doesn't look right and was wondering if someone could tell me if its a misprint or actually correct (and why)...
say you have <ψxψ⟩ or <0Fk⟩ where F is an operator, what does this actually mean? I understand Cψ⟩ would be the operator C acting on PSI and <ψ1ψ2⟩ is the inner product of two wavefunctions but what would a third term inbetween them mean?
thanks for any help
The question is to calculate the time evoution of S_{x} wrt <\Psi(t)\pm l where <\Psi\pm (t) l= ( \frac{1}{\sqrt{2}}(exp(^{+iwt})< \uparrow l , \pm exp(^{iwt})<
\downarrow l ) [1]
Sx=\frac{}{2}(^{0}_{1}^{1}_{0} )
Here is my attempt:
 First of all from [1] I see that l \Psi\pm (t) > = (...
1. Explain why <n(aa+)^3n> must be zero
2. a and a+ (a dagger) are the raising and lowering operators (creation and annihilation operators).
3. Because it says explain, I am not sure any mathematical proof is needed. I am best answer is that because (ignoring that the bracket...