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 f|v\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 anti-linear 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 bra-ket 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 bra-ket 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.)
Could someone please explain bra-ket notation in layman's terms?
Also could you please tell me how to notate this (bra-ket or otherwise)?
The probability of x_n is equal to \Lambda_n.
\Lambda_n is a value between 0 and 1.
x_n is, of course, position.
I have a basic question that I have overlooked in the past, given that you have
<psi2|A = lamda2<psi2|, where <| is a bra and lamda2 is the eigenvalue. If you were to multiply the equation by |psi1>, why do you get <psi2|A|psi1> = lamda2<psi2|psi1> and not |psi1><psi2|A = lamda2|psi1><psi2| ...
hey, can someone show me the step between these two lines of equations please:
(\Delta A)^2=<\psi|A^2|\psi>-<\psi|A|\psi>^2
=<\psi|(A-<A>)^2|\psi>
where A is an operator and \psi is the wavefunction and <A> is the expectation value of A
Homework Statement
my apologies if this question should be posted in the math forum
3-d space spanned by orthonormal basis: (kets) |1>, |2>, |3>. Ket |a> = i|1> - 2|2> - i|3>. Ket |b> = i|1> + 2|3>.
The question is to construct <a| and <b| in terms of the dual basis (kets 1,2,3)...
Hello.
I have been working out of the beginning of Shankar lately, and I wanted to address some confusion I have had with regard to Dirac notation.
I know that physicists tend to love the notation, but to me, so far, it is confusing, inconsistent, and even occasionally contradictory. Here...
I am reading a paper that uses a quantum mechanical notation that I do not understand. I have found a webpage that explains it, but I do not understand the explanation either:
http://chsfpc5.chem.ncsu.edu/~franzen/CH ...
(let Y represent the Psi)
Specifically, I understand what <Yn|A|Yn>...
I began my physics study about one year ago and learned all of classical mechanics in that year; but I am now studying Quantum Mechanics. The book I'm using (Dirac's Principles of Quantum Mechanics) Introduces Bra-Ket notation in the first chapter rather concisely. I understand the mathematical...
I need to show that (XY)^(dagger)=(Y)^(dagger)(X)^(dagger) using bra-ket algebra
where X and Y are operators
say we started out with: if we dagger it (using *):
XY|a> = X(Y|a>)=(X(Y|a>))=((Y|a>)X*)=<a|Y*X*
we also know that XY|a>=<a|(XY)* by definition, so...
I'm reading an article where there are an atom with two states, let's call them |0> and |1>. Then the writer defines an operator by
|0><1|
I know how this operator works in the bra ket notaion, but how does it work, if I want to use it in the position basis?
Someone told me that I just...
I'm supposed to perform the following operations.
|A + B> and <A + B|, where A and B are two vectors.
A = 3i |x> - 7i |y>
B = - |x> + 2i |y>
where |x> and |y> are orthonormal.
Little lost here... Is this asking me to add them component wise? ie
|A + B> = (3i - 1) |x> + (2i - 7i)...