States and Operators in Quantum Mechanics

Hilbert Space and Dirac Notation

Author

Daniel Fischer

Quantum Mechanics as Linear Algebra: Overview and Plan

In the previous chapter, we approached quantum mechanics primarily from the perspective of wave mechanics. Mathematically, this led us to differential equations: the wave function \(\psi(\vec r,t)\) evolves according to the Schrödinger equation, and stationary states are obtained by solving the time-independent Schrödinger equation.

There is, however, a more general mathematical structure underlying quantum mechanics.

Quantum states form a vector space, and physical quantities are represented by linear operators acting on this space. From this perspective, quantum mechanics is largely a theory of linear algebra.

This point of view is more general than the wave-mechanical description. A scalar spatial wave function \(\psi(\vec r)\) is only one possible representation of a quantum state. For example, the spin state of an electron cannot be described by a scalar function of position alone. More generally, quantum states may contain spatial, spin, internal, or other degrees of freedom.

The abstract state itself should therefore be distinguished from any particular representation of that state.

Dirac’s bra–ket notation provides a convenient language for doing this.

The specific goals for this chapter are:

  1. Introduce linear operators and identify familiar quantum-mechanical operators as examples.
  2. Recognize the Schrödinger equation and similar equations as eigenvalue equations.
  3. Understand why quantum states naturally form a complex vector space.
  4. Introduce the abstract state vector and Dirac’s bra–ket notation.
  5. Define inner products, norms, orthogonality, and complete bases in Hilbert space.
  6. Understand wave functions as the components of an abstract state in a particular basis.
  7. Represent linear operators through matrix elements, outer products, and projection operators.
  8. Introduce Hermitian operators and connect their eigenvalues and eigenstates to quantum measurements.
  9. Express measurement probabilities, expectation values, and uncertainties in Dirac notation.

States, bras, and inner products

A quantum state is represented abstractly by a ket

\[ \boxed{ \left\lvert \psi \right\rangle} \]

in a Hilbert space. The corresponding bra is

\[ \left\langle \psi \right\rvert. \]

A bra and a ket combine to form an inner product,

\[ \boxed{ \left\langle \phi \middle| \psi \right\rangle} \]

which is, in general, a complex number.

The inner product of a state with itself determines its norm,

\[ \|\psi\| = \sqrt{\left\langle \psi \middle| \psi \right\rangle}, \]

and a normalized quantum state satisfies

\[ \boxed{ \left\langle \psi \middle| \psi \right\rangle=1. } \]

Operators and eigenstates

A linear operator \(\hat A\) acts on states in Hilbert space. An eigenvalue equation has the form

\[ \boxed{ \hat A\left\lvert a \right\rangle=a\left\lvert a \right\rangle, } \]

where \(\left\lvert a \right\rangle\) is an eigenstate of \(\hat A\) and \(a\) is the corresponding eigenvalue.

Representations

The abstract state \(\left\lvert \psi \right\rangle\) can be represented in different bases.

In the position basis, the components of the state form the wave function,

\[ \boxed{ \psi(x)=\left\langle x \middle| \psi \right\rangle. } \]

Thus, the wave function is a representation of the abstract state rather than a different physical object.

Orthonormal bases and completeness

For an orthonormal basis \({\left\lvert n \right\rangle}\),

\[ \boxed{ \left\langle m \middle| n \right\rangle=\delta_{mn}. } \]

If the basis is complete,

\[ \boxed{ \sum_n \left\lvert n \right\rangle\!\!\left\langle n \right\rvert=\mathbb 1. } \]

Any state can therefore be expanded in the basis as

\[ \boxed{ \left\lvert \psi \right\rangle= \sum_n \left\lvert n \right\rangle\left\langle n \middle| \psi \right\rangle. } \]

The quantities

\[ \left\langle n \middle| \psi \right\rangle\]

are the components of the state in this basis.

Operators in a basis

Once a basis has been chosen, an operator is characterized by its matrix elements

\[ \boxed{ A_{mn} = \left\langle m \middle| \hat A \middle| n \right\rangle. } \]

Observables and measurement

Physical observables are represented by Hermitian operators,

\[ \boxed{ \hat A^\dagger=\hat A. } \]

Their eigenvalues are the possible measurement outcomes.

If the system is in the normalized state \(\left\lvert \psi \right\rangle\), the probability of obtaining the eigenvalue \(a_n\) is

\[ \boxed{ P(a_n) = \left| \left\langle a_n \middle| \psi \right\rangle\right|^2. } \]

The expectation value of the observable is

\[ \boxed{ \langle A\rangle = \left\langle \psi \middle| \hat A \middle| \psi \right\rangle, } \]

and its uncertainty is

\[ \boxed{ \sigma_A = \sqrt{ \langle A^2\rangle-\langle A\rangle^2 }. } \]

You should understand that

  • quantum states form a complex vector space, so linear combinations of states are again possible states;
  • the abstract state \(\left\lvert \psi \right\rangle\) is independent of how it is represented: \(\psi(x)=\left\langle x \middle| \psi \right\rangle\) and \(\tilde\psi(p)=\left\langle p \middle| \psi \right\rangle\) are different representations of the same state;
  • a bra–ket expression such as \(\left\langle \phi \middle| \psi \right\rangle\) is an inner product, while an outer product such as \(\left\lvert \phi \right\rangle\!\!\left\langle \psi \right\rvert\) is an operator;
  • choosing a basis is analogous to choosing coordinates for an ordinary vector; changing the basis changes the components, not the physical state;
  • the coefficients \(\left\langle n \middle| \psi \right\rangle\) are the projections of a state onto the basis states and determine its expansion in that basis;
  • the same abstract operator may appear as a differential operator in one representation and as a matrix in another;
  • eigenstates of an observable are states with definite values of that observable, while a general state is usually a superposition of several eigenstates;
  • for a Hermitian operator, eigenvalues are real and eigenstates belonging to different eigenvalues are orthogonal;
  • quantum measurement probabilities are obtained by projecting the state onto the eigenstates of the measured observable;
  • an expectation value is the statistical average of many measurements on identically prepared systems and need not itself be one of the possible measurement outcomes.

The detailed vector-space axioms, Riesz representation theorem, subtleties of generalized eigenvectors and rigged Hilbert spaces, and the distinction between Hermitian and self-adjoint operators are included for completeness but are not intended as material to memorize.

1. Operators and Linear Structure

1.1 Operators

An operator is a mathematical object that acts on another object and produces a new object.

For example,

\[ \hat A\psi=\phi \]

means that the operator \(\hat A\) acts on the function \(\psi\) and produces the function \(\phi\).

Several operators already appeared naturally in wave mechanics.

Physical quantity Operator in the position representation Action on \(\psi(x)\)
Position \(\hat x=x\) \(\hat x\psi(x)=x\psi(x)\)
Momentum \(\displaystyle \hat p_x=-i\hbar\frac{d}{dx}\) \(\displaystyle \hat p_x\psi=-i\hbar\frac{d\psi}{dx}\)
Potential energy \(\hat V=V(x)\) \(\hat V\psi=V(x)\psi\)
Total energy \(\displaystyle \hat H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\) \(\hat H\psi\)

The Hamiltonian \(\hat H\) is the operator associated with the total energy of the system.

For example, the time-independent Schrödinger equation is

\[ \hat H\psi(x)=E\psi(x). \]

In the time-dependent Schrödinger equation,

\[ i\hbar\frac{\partial}{\partial t}\psi = \hat H\psi, \]

the Hamiltonian generates the time evolution of the quantum state.

1.2 Linearity

The operators of quantum mechanics are linear operators.

An operator \(\hat A\) is linear if

\[ \boxed{ \hat A \left( c_1\psi_1+c_2\psi_2 \right) = c_1\hat A\psi_1 + c_2\hat A\psi_2 } \]

for arbitrary complex numbers \(c_1\) and \(c_2\).

As an example, consider the momentum operator:

\[ \hat p_x \left( c_1\psi_1+c_2\psi_2 \right) = -i\hbar \frac{d}{dx} \left( c_1\psi_1+c_2\psi_2 \right). \]

Using the linearity of differentiation,

\[ \hat p_x \left( c_1\psi_1+c_2\psi_2 \right) = c_1\hat p_x\psi_1+ c_2\hat p_x\psi_2. \]

Thus, \(\hat p_x\) is a linear operator.

The linearity of quantum-mechanical operators is closely connected to the superposition principle. If \(\psi_1\) and \(\psi_2\) are possible quantum states, then a linear combination

\[ \psi=c_1\psi_1+c_2\psi_2 \]

is also a possible state.

1.3 Eigenvalue equations

A particularly important situation occurs when the action of an operator reproduces the same function, multiplied only by a number:

\[ \boxed{ \hat A\psi=a\psi. } \]

This is an eigenvalue equation.

The function \(\psi\) is an eigenfunction of \(\hat A\), and \(a\) is the corresponding eigenvalue.

We have already encountered several examples.

A plane wave

\[ \psi_k(x)=e^{ikx} \]

satisfies

\[ \hat p_x\psi_k = -i\hbar\frac{d}{dx}e^{ikx} = \hbar k e^{ikx}. \]

Therefore,

\[ \hat p_x\psi_k=p\psi_k, \qquad p=\hbar k, \]

and the plane wave is an eigenfunction of the momentum operator.

Similarly, the time-independent Schrödinger equation,

\[ \hat H\psi_n=E_n\psi_n, \]

is an eigenvalue equation for the Hamiltonian.

The allowed energies \(E_n\) are therefore the eigenvalues of the Hamiltonian.

This observation suggests that the mathematical structure of quantum mechanics is closely related to the familiar eigenvalue problems of linear algebra.

2. Quantum States as Vectors

2.1 From functions to vectors

A wave function such as \(\psi(x)\) may not initially look like a vector.

We are accustomed to vectors written as columns,

\[ \vec v= \begin{pmatrix} v_1\\ v_2\\ v_3 \end{pmatrix}, \]

or represented geometrically as arrows.

But the defining feature of a vector is not that it has three components or that it can be drawn as an arrow.

What matters is that vectors can be added and multiplied by scalars while remaining elements of the same mathematical space.

Functions have precisely this property.

If \(\psi_1(x)\) and \(\psi_2(x)\) belong to an appropriate space of functions, then

\[ c_1\psi_1(x)+c_2\psi_2(x) \]

belongs to the same space.

Wave functions can therefore be regarded as vectors in a vector space.

A vector space \(V\) over the complex numbers \(\mathbb C\) is a set whose elements are called vectors, together with operations of vector addition and scalar multiplication.

For \(u,v,w\in V\) and \(a,b\in\mathbb C\), the following axioms hold:

  1. \(u+v\in V\);
  2. \(au\in V\);
  3. \(u+v=v+u\);
  4. \((u+v)+w=u+(v+w)\);
  5. there exists a zero vector \(0\) such that \(u+0=u\);
  6. for every \(u\) there exists \(-u\) such that \(u+(-u)=0\);
  7. \(a(u+v)=au+av\);
  8. \((a+b)u=au+bu\);
  9. \(a(bu)=(ab)u\);
  10. \(1u=u\).

Nothing in this definition requires a vector to be a column of numbers.

Vectors may be

  • ordinary geometric vectors,
  • columns of numbers,
  • matrices,
  • polynomials,
  • functions,
  • spinors,
  • or many other mathematical objects.

2.2 Hilbert space

The vector space used in quantum mechanics has additional structure.

In particular, we need to define an inner product between two vectors. This allows us to discuss lengths, angles, orthogonality, and projections.

A complex vector space equipped with an inner product and satisfying an additional mathematical completeness condition is called a Hilbert space, usually denoted by

\[ \mathscr H. \]

Quantum states are represented by vectors in a Hilbert space.

An inner product on a complex vector space assigns a complex number

\[ \langle\phi,\psi\rangle \]

to pairs of vectors.

It defines the norm

\[ \|\psi\| = \sqrt{\langle\psi,\psi\rangle}. \]

A Hilbert space is an inner-product space that is complete with respect to this norm: every Cauchy sequence of vectors converges to a vector that is also contained in the space — which is a fancy way of saying that you cannot get arbitrarily close to a perfectly good limit and then discover that the limit has somehow fallen outside the space.

For ordinary nonrelativistic wave mechanics, an important example is

\[ L^2(\mathbb R^3), \]

the space of square-integrable complex-valued functions.

The requirement

\[ \int_{\mathbb R^3}|\psi(\vec r)|^2\,d^3r<\infty \]

ensures that the norm is finite.

The normalized wave functions form only a subset of this vector space. They do not themselves form a vector space, because adding or rescaling them usually changes their normalization.

2.3 The abstract quantum state

Wave mechanics represents a state by a function such as

\[ \psi(\vec r). \]

But a spatial wave function is a particular representation of the state rather than the state itself.

For example, an electron may possess a spin state even when its spatial degree of freedom is irrelevant. A spin-\(1/2\) state may be represented by a two-component vector such as

\[ \begin{pmatrix} a\\ b \end{pmatrix}, \]

rather than by a scalar function of position.

Other quantum systems may be most naturally represented by occupation numbers, angular-momentum states, energy eigenstates, or other basis states.

It is therefore useful to introduce a notation for the quantum state that does not assume any particular representation.

Dirac introduced the notation

\[ \boxed{ \left\lvert \psi \right\rangle} \]

for such an abstract state vector.

The symbol \(\left\lvert \psi \right\rangle\) is called a ket.

3. Dirac Notation and the Dual Space

3.1 Ket vectors

A ket

\[ \left\lvert \psi \right\rangle\in\mathscr H \]

denotes a vector in the Hilbert space of quantum states.

Linear combinations of kets are again kets:

\[ \left\lvert \chi \right\rangle= c_1\left\lvert \psi \right\rangle+ c_2\left\lvert \phi \right\rangle. \]

The abstract eigenvalue equation is therefore written as

\[ \boxed{ \hat A\left\lvert a \right\rangle= a\left\lvert a \right\rangle. } \]

This equation makes no reference to coordinate space.

The same equation may later be represented as a differential equation, a matrix equation, or in some other basis.

Strictly speaking, a physical pure state is not represented by one unique vector \(\left\lvert \psi \right\rangle\).

The vectors

\[ \left\lvert \psi \right\rangle\qquad\text{and}\qquad e^{i\phi}\left\lvert \psi \right\rangle\]

differ only by an overall phase and represent the same physical state.

A physical pure state therefore corresponds mathematically to a ray in Hilbert space rather than to one particular vector.

In most calculations, however, we simply refer to \(\left\lvert \psi \right\rangle\) as the quantum state.

3.2 Bra vectors and the dual space

Associated with the Hilbert space \(\mathscr H\) is its dual space

\[ \mathscr H^\ast. \]

The dual space consists of linear functionals: maps from vectors in \(\mathscr H\) to complex numbers.

Associated with a ket \(\left\lvert \phi \right\rangle\) is a bra

\[ \left\langle \phi \right\rvert\in\mathscr H^\ast. \]

The bra acts on a ket according to

\[ \left\langle \phi \right\rvert: \left\lvert \psi \right\rangle\longmapsto \left\langle \phi \middle| \psi \right\rangle\in\mathbb C. \]

Thus,

\[ \boxed{ \left\langle \phi \middle| \psi \right\rangle} \]

is the inner product of the two states.

With the convention used in physics, the inner product is linear in the ket and conjugate-linear in the bra:

\[ \left\langle \phi \middle| c_1\psi_1+c_2\psi_2 \right\rangle= c_1\left\langle \phi \middle| \psi_1 \right\rangle+ c_2\left\langle \phi \middle| \psi_2 \right\rangle, \]

while

\[ \left( c_1\left\langle \phi_1 \right\rvert+ c_2\left\langle \phi_2 \right\rvert\right) \left\lvert \psi \right\rangle= c_1\left\langle \phi_1 \middle| \psi \right\rangle+ c_2\left\langle \phi_2 \middle| \psi \right\rangle, \]

but taking the bra associated with a linear combination of kets gives

\[ \left( c_1\left\lvert \phi_1 \right\rangle+ c_2\left\lvert \phi_2 \right\rangle\right)^\dagger = c_1^\ast\left\langle \phi_1 \right\rvert+ c_2^\ast\left\langle \phi_2 \right\rvert. \]

For a Hilbert space, the Riesz representation theorem states that every continuous linear functional can be represented uniquely by taking the inner product with an element of the Hilbert space.

This provides the mathematical connection between

\[ \left\lvert \psi \right\rangle\in\mathscr H \]

and

\[ \left\langle \psi \right\rvert\in\mathscr H^\ast. \]

Dirac notation hides much of this mathematical machinery, which is one reason it is so convenient.

3.3 Inner products, norms, and normalization

The inner product of a state with itself is

\[ \left\langle \psi \middle| \psi \right\rangle. \]

It is real and non-negative:

\[ \left\langle \psi \middle| \psi \right\rangle\geq0. \]

The norm of the state is

\[ \|\psi\| = \sqrt{\left\langle \psi \middle| \psi \right\rangle}. \]

A normalized state therefore satisfies

\[ \boxed{ \left\langle \psi \middle| \psi \right\rangle=1. } \]

4. Representations, Bases, and Orthogonality

4.1 The position representation

The abstract state \(\left\lvert \psi \right\rangle\) can be represented in a basis.

If we choose the position basis \(\{\left\lvert x \right\rangle\}\), the components of the state in this basis are

\[ \boxed{ \psi(x)=\left\langle x \middle| \psi \right\rangle. } \]

The familiar wave function is therefore the position-space representation of the abstract state.

Likewise, in the momentum basis,

\[ \boxed{ \tilde\psi(p)=\left\langle p \middle| \psi \right\rangle. } \]

These are not two different states.

They are two different representations of the same state \(\left\lvert \psi \right\rangle\).

This is analogous to representing one ordinary vector using two different coordinate systems.

4.2 Inner product in the position representation

For two states \(\left\lvert \phi \right\rangle\) and \(\left\lvert \psi \right\rangle\),

\[ \left\langle \phi \middle| \psi \right\rangle\]

is representation independent.

In the position representation it becomes

\[ \boxed{ \left\langle \phi \middle| \psi \right\rangle= \int_{-\infty}^{\infty} \phi^\ast(x)\psi(x)\,dx. } \]

Thus, the familiar integral over wave functions is not the fundamental definition of the quantum-mechanical inner product.

Rather, it is the form taken by the abstract inner product when the states are represented in the position basis.

4.3 Orthogonality

Two states are orthogonal if

\[ \boxed{ \left\langle \phi \middle| \psi \right\rangle=0. } \]

This is the generalization of perpendicular vectors in ordinary Euclidean space.

For ordinary vectors,

\[ \vec a\cdot\vec b=0 \]

indicates orthogonality.

For quantum states, the corresponding condition is

\[ \left\langle \phi \middle| \psi \right\rangle=0. \]

In the position representation,

\[ \int \phi^\ast(x)\psi(x)\,dx=0. \]

4.4 Orthonormal bases

Consider a discrete orthonormal set of states

\[ \{\left\lvert n \right\rangle\}. \]

Orthonormality means

\[ \boxed{ \left\langle m \middle| n \right\rangle=\delta_{mn}. } \]

If the basis is complete, any state can be expanded as

\[ \left\lvert \psi \right\rangle= \sum_n c_n\left\lvert n \right\rangle. \]

Taking the inner product with \(\left\langle m \right\rvert\) gives

\[ \left\langle m \middle| \psi \right\rangle= \sum_n c_n\left\langle m \middle| n \right\rangle= c_m. \]

Therefore,

\[ \boxed{ c_n=\left\langle n \middle| \psi \right\rangle. } \]

and

\[ \boxed{ \left\lvert \psi \right\rangle= \sum_n \left\lvert n \right\rangle\left\langle n \middle| \psi \right\rangle. } \]

4.5 Completeness

The previous expansion can be written as

\[ \left\lvert \psi \right\rangle= \left( \sum_n\left\lvert n \right\rangle\!\!\left\langle n \right\rvert\right) \left\lvert \psi \right\rangle. \]

Therefore, a complete orthonormal discrete basis satisfies

\[ \boxed{ \sum_n \left\lvert n \right\rangle\!\!\left\langle n \right\rvert= \mathbb 1. } \]

This is the completeness relation, or resolution of the identity.

For a continuous position basis,

\[ \boxed{ \int_{-\infty}^{\infty} \left\lvert x \right\rangle\!\!\left\langle x \right\rvert\,dx = \mathbb 1. } \]

The position eigenstates satisfy the delta-function normalization

\[ \left\langle x' \middle| x \right\rangle= \delta(x-x'). \]

There is a mathematical subtlety here.

Position and momentum eigenstates are not square-normalizable:

\[ \left\langle x \middle| x \right\rangle\]

and

\[ \left\langle p \middle| p \right\rangle\]

do not produce finite numbers.

They are therefore not ordinary elements of \(L^2\).

Dirac notation treats them as generalized eigenvectors, normalized using Dirac delta functions. A mathematically more rigorous treatment uses a rigged Hilbert space.

For our purposes, the usual Dirac notation is sufficient, but it is useful to remember that continuous-spectrum eigenstates require this extension of ordinary Hilbert-space language.

4.6 A linear-algebra dictionary

Abstract linear algebra Quantum mechanics in position space
State vector \(\left\lvert \psi \right\rangle\) Wave function \(\psi(x)=\left\langle x \middle| \psi \right\rangle\)
Inner product \(\left\langle \phi \middle| \psi \right\rangle\) \(\displaystyle \int\phi^\ast(x)\psi(x)\,dx\)
Basis vector \(\left\lvert n \right\rangle\) Basis function \(\phi_n(x)=\left\langle x \middle| n \right\rangle\)
Expansion coefficient \(\left\langle n \middle| \psi \right\rangle\) Projection of \(\psi(x)\) onto \(\phi_n(x)\)
Linear operator \(\hat A\) Matrix or differential operator
Matrix element \(\left\langle m \middle| \hat A \middle| n \right\rangle\) Integral involving basis functions and \(\hat A\)
Eigenvector \(\left\lvert a \right\rangle\) Eigenfunction of the corresponding differential operator

The central point is that the object \(\left\lvert \psi \right\rangle\) is independent of the basis.

The wave function \(\psi(x)\) is one particular set of coordinates used to describe it.

5. Linear Operators in Hilbert Space

5.1 Operators acting on kets

A linear operator maps vectors to vectors:

\[ \hat A:\mathscr H\rightarrow\mathscr H. \]

Thus,

\[ \hat A\left\lvert \psi \right\rangle= \left\lvert \phi \right\rangle. \]

More precisely, particularly for unbounded operators such as position and momentum, an operator may only be defined on a suitable domain

\[ D(\hat A)\subseteq\mathscr H. \]

We will normally leave the domain implicit.

The eigenvalue equation becomes

\[ \boxed{ \hat A\left\lvert a_n \right\rangle= a_n\left\lvert a_n \right\rangle. } \]

5.2 Matrix elements

Once a basis \(\{\left\lvert n \right\rangle\}\) has been selected, the operator can be represented by a matrix.

Its matrix elements are

\[ \boxed{ A_{mn} = \left\langle m \middle| \hat A \middle| n \right\rangle. } \]

Using completeness,

\[ \hat A = \mathbb 1\hat A\mathbb 1, \]

we obtain

\[ \hat A = \sum_{m,n} \left\lvert m \right\rangle\left\langle m \middle| \hat A \middle| n \right\rangle\left\langle n \right\rvert. \]

Thus,

\[ \boxed{ \hat A = \sum_{m,n} A_{mn}\left\lvert m \right\rangle\!\!\left\langle n \right\rvert. } \]

A differential operator and a matrix can therefore represent the same abstract linear operator in different bases.

5.3 Outer products and projectors

An inner product produces a complex number,

\[ \left\langle \phi \middle| \psi \right\rangle\in\mathbb C, \]

whereas an outer product

\[ \left\lvert \phi \right\rangle\!\!\left\langle \psi \right\rvert\]

produces an operator.

Acting on another state,

\[ \left\lvert \phi \right\rangle\!\!\left\langle \psi \right\rvert\chi\rangle = \left\lvert \phi \right\rangle\left\langle \psi \middle| \chi \right\rangle. \]

An important example is

\[ \boxed{ \hat P_n = \left\lvert n \right\rangle\!\!\left\langle n \right\rvert. } \]

For a normalized state \(\left\lvert n \right\rangle\),

\[ \hat P_n\left\lvert \psi \right\rangle= \left\lvert n \right\rangle\left\langle n \middle| \psi \right\rangle. \]

Thus, \(\hat P_n\) extracts the component of \(\left\lvert \psi \right\rangle\) parallel to \(\left\lvert n \right\rangle\).

It is therefore called a projection operator.

6. Adjoint and Hermitian Operators

6.1 Hermitian conjugation

The Hermitian conjugate, or adjoint, of a vector or operator is denoted by

\[ \dagger. \]

For kets and bras,

\[ \left\lvert \psi \right\rangle^\dagger = \left\langle \psi \right\rvert, \]

and

\[ \left\langle \psi \right\rvert^\dagger = \left\lvert \psi \right\rangle. \]

For complex numbers,

\[ c^\dagger=c^\ast. \]

For products, the order is reversed:

\[ (\hat A\hat B)^\dagger = \hat B^\dagger\hat A^\dagger. \]

For example,

\[ \left( c_1\left\lvert \psi_1 \right\rangle+ c_2\left\lvert \psi_2 \right\rangle\right)^\dagger = c_1^\ast\left\langle \psi_1 \right\rvert+ c_2^\ast\left\langle \psi_2 \right\rvert. \]

The inner product obeys

\[ \boxed{ \left\langle \phi \middle| \psi \right\rangle^\ast = \left\langle \psi \middle| \phi \right\rangle. } \]

For matrix elements,

\[ \boxed{ \left\langle \phi \middle| \hat A \middle| \psi \right\rangle^\ast = \left\langle \psi \middle| \hat A^\dagger \middle| \phi \right\rangle. } \]

6.2 Hermitian operators

An operator is called Hermitian if

\[ \boxed{ \hat A^\dagger=\hat A. } \]

Hermitian operators have real eigenvalues.

Let

\[ \hat A\left\lvert a \right\rangle=a\left\lvert a \right\rangle. \]

Then

\[ \left\langle a \middle| \hat A \middle| a \right\rangle= a\left\langle a \middle| a \right\rangle. \]

Taking the complex conjugate gives

\[ \left\langle a \middle| \hat A \middle| a \right\rangle^\ast = a^\ast\left\langle a \middle| a \right\rangle. \]

But

\[ \left\langle a \middle| \hat A \middle| a \right\rangle^\ast = \left\langle a \middle| \hat A^\dagger \middle| a \right\rangle. \]

For a Hermitian operator,

\[ \hat A^\dagger=\hat A, \]

so

\[ a\left\langle a \middle| a \right\rangle= a^\ast\left\langle a \middle| a \right\rangle. \]

For a nonzero eigenstate, \(\left\langle a \middle| a \right\rangle>0\), and therefore

\[ \boxed{a=a^\ast.} \]

Thus, \(a\) is real.

Suppose

\[ \hat A\left\lvert a \right\rangle=a\left\lvert a \right\rangle. \]

Then

\[ a\in\mathbb R. \]

Furthermore, eigenstates corresponding to distinct eigenvalues are orthogonal:

\[ a_m\neq a_n \quad\Rightarrow\quad \left\langle a_m \middle| a_n \right\rangle=0. \]

Let \(\left\lvert a_m \right\rangle\) and \(\left\lvert a_n \right\rangle\) be eigenstates of the Hermitian operator \(\hat A\) with different eigenvalues:

\[ \hat A\left\lvert a_m \right\rangle=a_m\left\lvert a_m \right\rangle, \]

and

\[ \hat A\left\lvert a_n \right\rangle=a_n\left\lvert a_n \right\rangle. \]

Consider the matrix element

\[ \left\langle a_m \middle| \hat A \middle| a_n \right\rangle. \]

Using the eigenvalue equation for \(\left\lvert a_n \right\rangle\),

\[ \left\langle a_m \middle| \hat A \middle| a_n \right\rangle= a_n\left\langle a_m \middle| a_n \right\rangle. \]

Because \(\hat A\) is Hermitian,

\[ \hat A^\dagger=\hat A, \]

we can also let the operator act to the left:

\[ \left\langle a_m \middle| \hat A \middle| a_n \right\rangle= a_m^\ast\left\langle a_m \middle| a_n \right\rangle. \]

The eigenvalues of a Hermitian operator are real, so

\[ a_m^\ast=a_m. \]

Therefore,

\[ a_n\left\langle a_m \middle| a_n \right\rangle= a_m\left\langle a_m \middle| a_n \right\rangle. \]

or

\[ (a_n-a_m)\left\langle a_m \middle| a_n \right\rangle=0. \]

If the eigenvalues are different,

\[ a_n\neq a_m, \]

the only possibility is

\[ \boxed{ \left\langle a_m \middle| a_n \right\rangle=0. } \]

Thus, eigenstates of a Hermitian operator belonging to different eigenvalues are orthogonal.

In introductory quantum mechanics, observables are usually said to be represented by Hermitian operators, satisfying

\[ \hat A^\dagger=\hat A. \]

For finite-dimensional matrices, this terminology is unambiguous.

For unbounded operators in infinite-dimensional Hilbert spaces, such as position and momentum, there is a technical distinction between a symmetric operator and a fully self-adjoint operator. The distinction involves the domains on which the operators are defined.

Mathematically, quantum-mechanical observables are more precisely represented by self-adjoint operators. In these notes, we will follow the usual physics convention and use the term Hermitian operator.

7. Observables and Measurement

7.1 Observables

A measurable physical quantity is represented by a Hermitian—or, more precisely, self-adjoint—operator.

Examples include

  • position \(\hat x\),
  • momentum \(\hat p\),
  • energy \(\hat H\),
  • angular momentum \(\hat{\mathbf L}\),
  • spin \(\hat{\mathbf S}\).

Possible outcomes of a measurement are given by the spectrum of the corresponding operator.

For a discrete spectrum,

\[ \hat A\left\lvert a_n \right\rangle= a_n\left\lvert a_n \right\rangle. \]

The values \(a_n\) are possible measurement outcomes.

7.2 Spectral decomposition

For a nondegenerate discrete spectrum with a complete orthonormal set of eigenvectors,

\[ \boxed{ \hat A = \sum_n a_n\left\lvert a_n \right\rangle\!\!\left\langle a_n \right\rvert. } \]

This expression is the operator analogue of diagonalizing a matrix.

If degeneracies are present, the projection operator onto each eigenspace must include all states belonging to that eigenvalue.

Continuous spectra require the corresponding integral form.

7.3 Measurement probabilities

Suppose the system is in the normalized state

\[ \left\lvert \psi \right\rangle. \]

Expanding it in eigenstates of \(\hat A\) gives

\[ \left\lvert \psi \right\rangle= \sum_n c_n\left\lvert a_n \right\rangle, \]

with

\[ c_n = \left\langle a_n \middle| \psi \right\rangle. \]

The Born rule states that the probability of obtaining the result \(a_n\) is

\[ \boxed{ P(a_n) = \left| \left\langle a_n \middle| \psi \right\rangle\right|^2. } \]

Thus, measurement probabilities are determined by the projection of the state vector onto the eigenvectors of the observable.

7.4 Expectation values

The expectation value of an observable \(\hat A\) in the state \(\left\lvert \psi \right\rangle\) is

\[ \boxed{ \langle A\rangle = \left\langle \psi \middle| \hat A \middle| \psi \right\rangle. } \]

In the position representation,

\[ \langle A\rangle = \int \psi^\ast(x) \hat A \psi(x)\,dx. \]

For a state expanded in the eigenbasis of \(\hat A\),

\[ \left\lvert \psi \right\rangle= \sum_n c_n\left\lvert a_n \right\rangle, \]

the expectation value becomes

\[ \langle A\rangle = \sum_n |c_n|^2a_n. \]

Thus, the abstract operator expression reproduces the usual statistical average over possible measurement results.

7.5 Variance and uncertainty

The variance of an observable is

\[ \boxed{ \sigma_A^2 = \left\langle (\hat A-\langle A\rangle)^2 \right\rangle. } \]

Expanding this expression gives

\[ \boxed{ \sigma_A^2 = \langle A^2\rangle - \langle A\rangle^2. } \]

The corresponding standard deviation is

\[ \boxed{ \sigma_A = \sqrt{ \langle A^2\rangle - \langle A\rangle^2 }. } \]

This quantity measures the spread of the possible outcomes of repeated measurements of \(A\).

In the next chapter, we will see that the uncertainties of two observables can be fundamentally related when the corresponding operators do not commute.

8. The Central Idea

The wave-mechanical and linear-algebra descriptions of quantum mechanics are not competing theories.

They are two ways of describing the same physics.

The wave function

\[ \psi(x) \]

is a representation of the abstract state

\[ \left\lvert \psi \right\rangle\]

in the position basis:

\[ \boxed{ \psi(x)=\left\langle x \middle| \psi \right\rangle. } \]

Similarly,

\[ \tilde\psi(p)=\left\langle p \middle| \psi \right\rangle\]

is the representation of the same state in the momentum basis.

The abstract Hilbert-space formulation is therefore more general than any one particular wave-function representation.

It allows us to discuss spatial wave functions, spin states, angular-momentum states, atomic levels, and many other quantum systems using the same mathematical language:

\[ \boxed{ \text{states are vectors, and observables are operators.} } \]