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Bra-ket notation
1
Bra-ket notation
In quantum mechanics, Bra-ket notation is a standard notation for describing quantum states, composed of angle
brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It is so
called because the inner product (or dot product) of two states is denoted by a ⟨bra|c|ket⟩;
,
consisting of a left part,
, called the bra (
/brɑː/), and a right part,
, called the ket (
/kɛt/). The
[1]
notation was introduced in 1939 by Paul Dirac and is also known as Dirac notation, though the notation has
precursors in Grassmann's use of the notation
for his inner products nearly 100 years previously.[2]
Bra-ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum
mechanics—including a large portion of modern physics — is usually explained with the help of bra-ket notation.
The expression
is typically interpreted as the probability amplitude for the state ψ to collapse into the state
ϕ.
Vector spaces
Background: Vector spaces
In physics, basis vectors allow any vector to be represented geometrically using angles and lengths, in different
directions, i.e. in terms of the spatial orientations. It is simpler to see the notational equivalences between ordinary
notation and bra-ket notation, so for now; consider a vector A as an element of 3-d Euclidean space using the field of
real numbers, symbolically stated as
.
The vector A can be written using any set of basis vectors and corresponding coordinate system. Informally basis
vectors are like "building blocks of a vector", they are added together to make a vector, and the coordinates are the
number of basis vectors in each direction. Two useful representations of a vector are simply a linear combination of
basis vectors, and column matrices. Using the familiar cartesian basis, a vector A is written;
Illustration of cartesian vectors, bases, coordinates and components. The coordinates of
the vector are equal to the projections of the vector (yellow) onto the x-component basis
vector (green) - using the dot product (a special case of an inner product, see below).
Bra-ket notation
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respectively, where ex, ey, ez denotes the cartesian basis vectors (all are orthogonal unit vectors) and Ax, Ay, Az are the
corresponding coordinates, in the x, y, z directions. In a more general notation, for any basis in 3d space we write;
Generalizing further, consider a vector A in an N dimensional vector space over the field of complex numbers
symbolically stated as
,
. The vector A is still conventionally represented by a linear combination of basis
vectors or a column matrix:
though the coordinates and vectors are now all complex-valued.
Even more generally, A can be a vector in a complex Hilbert space. Some Hilbert spaces, like
, have finite
dimension, while others have infinite dimension. In an infinite-dimensional space, the column-vector representation
of A would be a list of infinitely many complex numbers.
Bra-ket notation
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Ket notation for vectors
Rather than boldtype, over/under-arrows, underscores etc. conventionally used elsewhere;
notation for a vector uses vertical bars and angular brackets;
[3]
called ket, read as "ket-A".
, Dirac's
. When this notation is used, these vectors are
This applies to all vectors, the resultant vector and the basis. The previous vectors are
now written
or in a more easily generalized
notation,
The last one may be written in short as
Notice how any symbols, letters,
numbers, or even words — whatever
serves as a convenient label — can be
used as the label inside a ket. In other
words, the symbol " " has a
specific and universal mathematical
Illustration of cartesian vectors, bases, coordinates and components. The coordinates of
the
vector are equal to the projections of the vector (yellow) onto the x-component basis
meaning, but just the "A" by itself does
vector (green) - using the inner product (see below).
not. Nevertheless, for convenience,
there is usually some logical scheme
behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics with a
list of their quantum numbers.
Inner products and bras
An inner product is a generalization of the dot product. The inner product of two vectors is a complex number.
Bra-ket notation uses a specific notation for inner products:
For example, in three-dimensional complex Euclidean space,
where
denotes the complex conjugate of
. A special case is the inner product of a vector with itself, which is
square of its norm (magnitude):
Bra-ket notation splits this inner product (also called a "bracket") into two pieces, the "bra" and the "ket":
where
is called a bra, read as "bra-A", and
is a ket as above.
The purpose of "splitting" the inner product into a bra and a ket is that both the bra
and the ket
are
meaningful on their own, and can be used in other contexts besides within an inner product. There are two main
ways to think about the meanings of separate bras and kets:
Bra-ket notation
Bras and kets as row and column vectors
For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix
multiplication of a row vector with a column vector:
Based on this, the bras and kets can be defined as:
and then it is understood that a bra next to a ket implies matrix multiplication.
The conjugate transpose (also called Hermitian conjugate) of a bra is the corresponding ket and vice-versa:
because if one starts with the bra
then performs a complex conjugation, and then a matrix transpose, one ends up with the ket
Bras as linear operators on kets
A more abstract definition, which is equivalent but more easily generalized to infinite-dimensional spaces, is to say
that bras are linear functionals on kets, i.e. operators that input a ket and output a complex number. The bra operators
are defined to be consistent with the inner product.
In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding
bras and kets are related by the Riesz representation theorem.
Non-normalizable states and non-Hilbert spaces
Bra-ket notation can be used even if the vector space is not a Hilbert space.
In quantum mechanics, it is common practice to write down kets which have infinite norm, i.e. non-normalisable
wavefunctions. Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves.
These do not, technically, belong to the Hilbert space itself. However, the definition of "Hilbert space" can be
broadened to accommodate these states (see the Gelfand–Naimark–Segal construction or rigged Hilbert spaces). The
bra-ket notation continues to work in an analogous way in this broader context.
For a rigorous treatment of the Dirac inner product of non-normalizable states see the definition given by D. Carfì in
[4]
and.[4] For a rigorous definition of basis with a continuous set of indices and consequently for a rigorous
definition of position and momentum basis see.[4] For a rigorous statement of the expansion of an S-diagonalizable
operator - observable - in its eigenbasis or in another basis see.[4]
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Bra-ket notation
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Banach spaces are a different generalization of Hilbert spaces. In a Banach space B, the vectors may be notated by
kets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate the
vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the
meaning of an inner product, because the Riesz representation theorem does not apply.
Usage in quantum mechanics
The mathematical structure of quantum mechanics is based in large part on linear algebra:
• Wave functions and other quantum states can be represented as vectors in a complex Hilbert space. (The exact
structure of this Hilbert space depends on the situation.) In bra-ket notation, for example, an electron might be in
"the state
". (Technically, the quantum states are rays of vectors in the Hilbert space, as
corresponds
to the same state for any nonzero complex number c.)
• Quantum superpositions can be described as vector sums of the constituent states. For example, an electron in the
state
is in a quantum superposition of the states
and
.
• Measurements are associated with linear operators (called observables) on the Hilbert space of quantum states.
• Dynamics is also described by linear operators on the Hilbert space. For example, in the Schrödinger picture,
there is a linear operator U with the property that if an electron is in state
right now, then in one minute it
will be in the state
, the same U for every possible
.
• Wave function normalization is scaling a wave function so that its norm is 1.
Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and
often does involve, bra-ket notation. A few examples follow:
Position-space wave function
The Hilbert space of a spin-0 point particle is spanned by a "position basis"
, where the label r extends over
the set of all points in space. Since there are infinitely many vectors in the basis, this is an infinite-dimensional
Hilbert space.
Starting from any ket
in this Hilbert space, we can define a complex scalar function of r, known as a
wavefunction:
On the left side,
is a function mapping any point in space to a complex number; on the right side,
is a ket.
It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by
For instance, the momentum operator p has the following form:
One occasionally encounters an expression like
though this is something of a (fairly common) abuse of notation. The differential operator must be understood to be
an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is
projected into the position basis:
Bra-ket notation
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Overlap of states
In quantum mechanics the expression
collapse into the state
is typically interpreted as the probability amplitude for the state
. Mathematically, this means the coefficient for the projection of
onto
to
.
Changing basis for a spin-1/2 particle
A stationary spin-½ particle has a two-dimensional Hilbert space. One orthonormal basis is:
where
is the state with a definite value of the spin operator Sz equal to +1/2 and
is the state with a
definite value of the spin operator Sz equal to -1/2.
Since these are a basis, any quantum state of the particle can be expressed as a linear combination (i.e., quantum
superposition) of these two states:
where
are complex numbers.
A different basis for the same Hilbert space is:
defined in terms of Sx rather than Sz.
Again, any state of the particle can be expressed as a linear combination of these two:
In vector form, you might write
depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used.
There is a mathematical relationship between
; see change of basis.
Misleading uses
There are few conventions and notation abuses generally accepted on the physical community which might confuse
the non initiated.
It is common among physicists to use the same symbol for labels and constants on the same equation. It supposedly
becomes easier to identify that the constant is related to the labeled object, and is claimed that the divergent nature of
each will eliminate any ambiguity and no further differentiation is required. A common example is the usual
definition of a quantum coherent state
as satisfying
where is both the name of the state and
its eigenvalue(a complex number) with respect to the lowering operator.
Something similar occurs in component notation of vectors. While
wavefunctions,
(uppercase) is traditionally associated with
(lowercase) may be used to denote a label, a wave function or complex constant in the same
context, usually differentiated only by a subscript.
The main abuses are including operations inside the vector labels. This is usually done for a fast notation of scaling
vectors. E.g. if the vector
is scaled by
, it might be denoted by
, which makes no sense since
is a label, not a function or a number, so you can't make operations on it.
This is specially common when denoting vectors as tensor products, where part of the labels are moved outside the
designed slot. E.g.
. Here part of the labeling that should state that all three vectors
are different was moved outside the kets, as subscripts 1 and 2. And a further abuse occurs, since
to the norm of the first vector - which is a label is denoting a value.
is meant to refer
Bra-ket notation
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Linear operators
Linear operators acting on kets
A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have
certain properties.) In other words, if A is a linear operator and
is a ket, then
is another ket.
In an N-dimensional Hilbert space,
can be written as an N×1 column vector, and then A is an N×N matrix with
complex entries. The ket
can be computed by normal matrix multiplication.
Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are
represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are
represented by unitary linear operators such as rotation or the progression of time.
Linear operators acting on bras
Operators can also be viewed as acting on bras from the right hand side. Specifically, if A is a linear operator and
is a bra, then
is another bra defined by the rule
.
(in other words, a function composition). This expression is commonly written as (cf. energy inner product)
In an N-dimensional Hilbert space,
can be written as a 1×N row vector, and A (as in the previous section) is an
N×N matrix. Then the bra
can be computed by normal matrix multiplication.
If the same state vector appears on both bra and ket side,
then this expression gives the expectation value, or mean or average value, of the observable represented by operator
A for the physical system in the state
.
Outer products
A convenient way to define linear operators on H is given by the outer product: if
is a bra and
is a ket, the
outer product
denotes the rank-one operator that maps the ket
to the ket
(where
is a scalar multiplying
the vector
).
For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:
The outer product is an N×N matrix, as expected for a linear operator.
One of the uses of the outer product is to construct projection operators. Given a ket
projection onto the subspace spanned by
is
of norm 1, the orthogonal
Bra-ket notation
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Hermitian conjugate operator
Just as kets and bras can be transformed into each other (making
corresponding with
is
into
) the element from the dual space
where A† denotes the Hermitian conjugate (or adjoint) of the operator A. In
other words,
if and only if
.
†
If A is expressed as an N×N matrix, then A is its conjugate transpose.
Self-adjoint operators, where A=A†, play an important role in quantum mechanics; for example, an observable is
always described by a self-adjoint operator. If A is a self-adjoint operator, then
is always a real number
(not complex). This implies that expectation values of observables are real.
Properties
Bra-ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the
properties that allow this manipulation are listed herein. In what follows, c1 and c2 denote arbitrary complex
numbers, c* denotes the complex conjugate of c, A and B denote arbitrary linear operators, and these properties are
to hold for any choice of bras and kets.
Linearity
• Since bras are linear functionals,
• By the definition of addition and scalar multiplication of linear functionals in the dual space,[5]
Associativity
Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators
(but not addition), written in bra-ket notation, the parenthetical groupings do not matter (i.e., the associative property
holds). For example:
and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously
because of the equalities on the left. Note that the associative property does not hold for expressions that include
non-linear operators, such as the antilinear time reversal operator in physics.
Hermitian conjugation
Bra-ket notation makes it particularly easy to compute the Hermitian conjugate (also called dagger, and denoted †)
of expressions. The formal rules are:
• The Hermitian conjugate of a bra is the corresponding ket, and vice-versa.
• The Hermitian conjugate of a complex number is its complex conjugate.
• The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is
itself—i.e.,
.
Bra-ket notation
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• Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators,
written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components,
and taking the Hermitian conjugate of each.
These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as
follows:
• Kets:
• Inner products:
• Matrix elements:
• Outer products:
Composite bras and kets
Two Hilbert spaces V and W may form a third space
by a tensor product. In quantum mechanics, this is
used for describing composite systems. If a system is composed of two subsystems described in V and W
respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to
this is if the subsystems are actually identical particles. In that case, the situation is a little more complicated.)
If
is a ket in V and
is a ket in W, the direct product of the two kets is a ket in
. This is written
variously as
or
or
or
The unit operator
Consider a complete orthonormal system (basis),
from an inner product
with
, for a Hilbert space H, with respect to the norm
. From basic functional analysis we know that any ket
can be written as
the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars now
follows that
must be the unit operator, which sends each vector to itself. This can be inserted in any expression without affecting
its value, for example
where in the last identity Einstein summation convention has been used.
In quantum mechanics it often occurs that little or no information about the inner product
of two arbitrary
(state) kets is present, while it is possible to say something about the expansion coefficients
and
of those vectors with respect to a chosen (orthonormalized) basis. In this case it is particularly useful to
insert the unit operator into the bracket one time or more (for more information see Resolution of the identity).
Bra-ket notation
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Notation used by mathematicians
The object physicists are considering when using the "bra-ket" notation is a Hilbert space (a complete inner product
space).
Let
be a Hilbert space and
is a vector in
. What physicists would denote as
is the vector itself.
That is
.
Let
be the dual space of
is defined by
. This is the space of linear functionals on
where for all
. The isomorphism
we have
,
where
and
are just different notations for expressing an inner product between two
elements in a Hilbert space (or for the first three, in any inner product space). Notational confusion arises when
identifying
and with
and
respectively. This is because of literal symbolic substitutions. Let
and let
. This gives
One ignores the parentheses and removes the double bars. Some properties of this notation are convenient since we
are dealing with linear operators and composition acts like a ring multiplication.
Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second
one, and they don't use the *-symbol, but an overline (which the physicists reserve to averages) to denote
conjugate-complex numbers, i.e. for scalar products mathematicians usually write
whereas physicists would write for the same quantity
References and notes
[1] PAM Dirac (1939). "A new notation for quantum mechanics" (http:/ / journals. cambridge. org/ action/ displayAbstract?fromPage=online&
aid=2031476). Mathematical Proceedings of the Cambridge Philosophical Society 35 (3): pp. 416-418. doi:10.1017/S0305004100021162. .
[2] H. Grassmann (1862). Extension Theory. History of Mathematics Sources. American Mathematical Society, London Mathematical Society,
2000 translation by Lloyd C. Kannenberg.
[3] Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
[4] Carfì, David (April 2003). "Dirac-orthogonality in the space of tempered distributions" (http:/ / portal. acm. org/ citation. cfm?id=774918).
Journal of Computational and Applied Mathematics 153 (1-2): 99–107. Bibcode 2003JCoAM.153...99C.
doi:10.1016/S0377-0427(02)00634-9. .
[5] Lecture notes by Robert Littlejohn (http:/ / bohr. physics. berkeley. edu/ classes/ 221/ 1112/ notes/ hilbert. pdf), eqns 12 and 13
Bra-ket notation
Further reading
• Feynman, Leighton and Sands (1965). The Feynman Lectures on Physics Vol. III. Addison-Wesley.
ISBN 0-201-02115-3.
External links
• Richard Fitzpatrick, "Quantum Mechanics: A graduate level course" (http://farside.ph.utexas.edu/teaching/
qm/lectures/lectures.html), The University of Texas at Austin.
• 1. Ket space (http://farside.ph.utexas.edu/teaching/qm/lectures/node7.html)
• 2. Bra space (http://farside.ph.utexas.edu/teaching/qm/lectures/node8.html)
• 3. Operators (http://farside.ph.utexas.edu/teaching/qm/lectures/node9.html)
• 4. The outer product (http://farside.ph.utexas.edu/teaching/qm/lectures/node10.html)
• 5. Eigenvalues and eigenvectors (http://farside.ph.utexas.edu/teaching/qm/lectures/node11.html)
• Robert Littlejohn, Lecture notes on "The Mathematical Formalism of Quantum mechanics", including bra-ket
notation. (http://bohr.physics.berkeley.edu/classes/221/0708/notes/hilbert.pdf)
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