Linear Vector Spaces and Operators

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle} evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

Quantum mechanics can be conveniently formulated in the language of abstract state vectors, from which the various representations (wave mechanics, matrix mechanics, Schrödinger, Heisenberg and interaction pictures, etc.) can be derived. A formulation of quantum mechanics in terms of linear vector spaces hinges on the fact that the Schrödinger equation is linear. An operator defines a mathematical operation performed on a vector belonging to a linear vector space, the result of which is another vector belonging to the same linear vector space.

The Vector (Ket) Space

In quantum mechanics, a physical state is represented by a state vector in a complex linear vector space. Following Dirac, we call such a vector a "ket", denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle.} This state vector is postulated to contain complete information about the physical state (i.e. everything we are allowed to ask about the state is contained in the vector). The complex linear vector spaces that we work with in quantum mechanics are usually infinite dimensional. In this case, the vector space in question is known as a Hilbert space after D. Hilbert, who studied vector spaces in infinite dimensions.

One of the postulates of quantum mechanics is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c|\alpha\rangle,} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\neq 0,} represent the same physical state. In other words, only the ”direction” in vector space is of significance.

Since we assume that these vectors belong to a linear vector space, we may, given any set of state vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle,|\beta\rangle,\ldots,} form a superposition of the states, given by a linear combination of the vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{\alpha}|\alpha\rangle+c_{\beta}|\beta\rangle+\cdots.}

The Dual (Bra) Space

The vector space we have been dealing with is a ket space. We now introduce the notion of a "bra" space, a vector space ”dual to” the ket space. We postulate that corresponding to every ket Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle} there exists a bra denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\alpha |} in this dual or bra space. The dual space is spanned by a set of bra vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left \{\langle\alpha |\right \}} which correspond to the set of kets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left \{|\alpha\rangle\right \}.} Mathematically, the dual space is a set of linear functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\beta|: V \rightarrow C } that act on the members of the corresponding vector space where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\!} is the vector space and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\!} is the set of complex numbers.

There is a one-to-one correspondence between the members of a ket space and those of the corresponding bra space, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha\rangle\leftrightarrow\langle\alpha |,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leftrightarrow} stands for dual correspondence. Roughly speaking, we can regard the bra space as some kind of "mirror image" of the ket space.

The bra dual to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c|\alpha\rangle} is postulated to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^{\ast}\langle\alpha |} , not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\langle\alpha |,} which is a very important point to note. More generally, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{\alpha}|\alpha\rangle+c_{\beta}|\beta\rangle+\cdots\leftrightarrow c^{\ast}_{\alpha}\langle\alpha |+c^{\ast}_{\beta}\langle\beta |+\cdots}

The Hilbert Space

A Hilbert space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\!} consisting of a set of vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi,\phi,\chi,\ldots\!} and a set of scalars Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c,\ldots\!} obeys the following properties.

(a) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\!} is a linear vector space. It obeys all the properties of a linear vector space as mentioned in the previous section.

(b) The scalar product defined in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\!} is strictly positive. The scalar product of one element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi\!} with another element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\!} is a complex number, denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\phi|\psi\rangle} . This scalar satisfies the following properties.

(1) The scalar product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi\!} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\!} is same as the complex conjugate of the scalar product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\!} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi.\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\phi|\psi\rangle=\langle\psi|\phi\rangle^{\ast}}

(2) The scalar product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\!} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi\!} is linear with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi.\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi|a\psi_{1} + b\psi_{2}\rangle=a\langle\psi|\phi_{1}\rangle + b\langle\psi|\phi_{2}\rangle}

(3) The scalar product of a state vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\!} with itself is a positive real number.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi|\psi\rangle=|\psi|^2\geq 0}

In terms of this scalar product, we may express the normalization of a wave function as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi|\psi\rangle=1.}

Schwartz Inequality

For two states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle} belonging to a linear vector space, the following theorem, known as the Schwartz inequality, holds:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\langle\psi|\phi\rangle|^{2}\leq\langle\psi|\psi\rangle\langle\phi|\phi\rangle}

If the vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle} are linearly dependent, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle=\alpha|\phi\rangle,} then the above relation becomes an equality.

Proof: Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle} be arbitrary vectors in the vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V.} The inequality is trivial in the case that at least one of $|\psi \rangle$ and $|\phi \rangle =0,$ so we will consider the case that both are nonzero. Let $\lambda$ be a complex number. Then

$0\leq \langle \psi -\lambda \phi |\psi -\lambda \phi \rangle =\langle \psi |\psi \rangle -\lambda ^{\ast }\langle \psi |\phi \rangle -\lambda \langle \phi |\psi \rangle +|\lambda |^{2}\langle \phi |\phi \rangle .$

The above expression is valid for any value of $\lambda$. The right-hand side of the above expression is minimized if we choose

$\lambda ={\frac {\langle \psi |\phi \rangle }{\langle \phi |\phi \rangle }}.$

Using this value of $\lambda ,$ we obtain

$0\leq \langle \psi |\psi \rangle -{\frac {|\langle \psi |\phi \rangle |^{2}}{\langle \phi |\phi \rangle }},$

or

$|\langle \psi |\phi \rangle |^{2}\leq \langle \psi |\psi \rangle \langle \phi |\phi \rangle .$

Linear Operators

Let $V$ be a linear vector space. A linear operator ${\hat {A}}$ is an operation, denoted by ${\hat {A}}|\psi \rangle ,$ that maps a given ket vector $|\psi \rangle$ in $V$ to a different vector $|{\hat {A}}\psi \rangle$ in the same space, and has the property that ${\hat {A}}\left(c_{\psi }|\psi \rangle +c_{\phi }|\phi \rangle \right)=c_{\psi }{\hat {A}}|\psi \rangle +c_{\phi }{\hat {A}}|\phi \rangle .$

In addition, linear operators obey the following properties.

(1) If ${\hat {A}}|\psi \rangle ={\hat {B}}|\psi \rangle$ for every $|\psi \rangle ,$ then ${\hat {A}}$ is equal to ${\hat {B}}$.

(2) Commutative law: ${\hat {A}}+{\hat {B}}={\hat {B}}+{\hat {A}}\!$

(3) Associative law: $({\hat {A}}+{\hat {B}})+{\hat {C}}={\hat {A}}+({\hat {B}}+{\hat {C}})\!$

(4) Multiplication of operators: ${\hat {A}}{\hat {B}}|\psi \rangle ={\hat {A}}({\hat {B}}|\psi \rangle )$

(5) There exists an identity operator ${\hat {I}}$ such that ${\hat {I}}|\psi \rangle =|\psi \rangle .$

For some, but not all, operators ${\hat {A}},$ there exists an inverse operator ${\hat {A}}^{-1}$ such that ${\hat {A}}^{-1}{\hat {A}}={\hat {A}}{\hat {A}}^{-1}={\hat {I}}.$

The Hermitian Adjoint

The dual vector to ${\hat {A}}|\psi \rangle$, where ${\hat {A}}$ is a linear operator, is $\langle {\hat {A}}\psi |=\langle \psi |{\hat {A}}^{\dagger },$ where ${\hat {A}}^{\dagger }$ is known as the Hermitian adjoint of ${\hat {A}},$ and is itself a linear operator. The properties obeyed by Hermitian adjoints are as follows.

(1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\hat{A}^{\dagger})^{\dagger}=\hat{A}}
(2) For any complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c,\,(c\hat{A})^{\dagger}=c^{\ast}\hat{A}^{\dagger}}
(3) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\hat{A}+\hat{B})^{\dagger}=\hat{A}^{\dagger}+\hat{B}^{\dagger} }
(4) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\hat{A}\hat{B})^{\dagger}=\hat{B}^{\dagger}\hat{A}^{\dagger}}
(5) For any complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c,\,c^{\dagger}=c^{\ast}}
(6) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a\rangle ^{\dagger}=\langle a| }
(7) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (|a\rangle\langle b|)^{\dagger}=|b\rangle\langle a|}

Linear Independence and Bases

Consider Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\!} vectors (ket states) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1\rangle } ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |2\rangle,\ldots,|N\rangle} belonging to a linear vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V.} They are linearly independent if the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^{N}c_i|i\rangle=0 } necessarily implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i=0 \!} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 1,2,...,N\!} . They can be used as a basis in a vector space, and decomposition of any vector in terms of basis vectors in unique. Any such set of basis vectors must be complete; i.e., any vector can be written as a linear combination of vectors from the set.

While any set of linearly independent vectors can be used as a basis, it is usually easier to work in an orthonormal basis; i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle i|j \rangle=\delta_{ij}.} For such a case, we may write a completeness relation for the basis. Consider an arbitrary vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} expanded in terms of a given orthonormal basis:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle=\sum_{i=1}^{N}c_i|i\rangle}

We may find the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i\!} by simply taking the scalar product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} with each of the basis vectors, obtaining

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i=\langle i|\psi\rangle.}

Substituting back into the expansion for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle,} we find that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^{N}|i\rangle\langle i|\psi\rangle=|\psi\rangle.}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} is arbitrary, we conclude that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^{N}|i\rangle\langle i|=\hat{I},}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{I}} is the identity operator. This is the completeness relation that we sought. It is possible to derive similar relations for more general bases through a similar line of reasoning.

Matrix Elements of a Linear Operator

The action of a linear operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} is completely known once its action on each of the basis vectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is given. To see this, let us consider the action of such an operator on an arbitrary vector:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}|\psi\rangle=\sum_{i=1}^{N}c_i\hat{A}|i\rangle=\sum_{i=1}^{N}\sum_{j=1}^{N}c_i A_{ji}|j\rangle,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{ji}=\langle j|\hat{A}|i\rangle.} Substituting in the expression for the $c_{i}\!$ derived earlier, we obtain

${\hat {A}}|\psi \rangle =\sum _{i=1}^{N}\sum _{j=1}^{N}A_{ji}|j\rangle \langle i|\psi \rangle .$

Again, since $|\psi \rangle$ is arbitrary, we conclude that

${\hat {A}}=\sum _{i=1}^{N}\sum _{j=1}^{N}A_{ji}|j\rangle \langle i|.$

The coefficients $A_{ji}\!$ appearing in the above expression are known as the matrix elements of the operator ${\hat {A}}.$ This is because the expansion coefficients for the "input" and "output" vectors are related by a matrix equation, with the $A_{ji}\!$ being the elements of the matrix appearing in said equation. Let us write the "output" ${\hat {A}}|\psi \rangle$ as

${\hat {A}}|\psi \rangle =\sum _{j=1}^{N}c'_{j}|j\rangle .$

We see, however, from our earlier derivation that

$c'_{j}=\sum _{i=1}^{N}A_{ji}c_{i},$

or $\mathbf {c} '=\mathbf {A} \mathbf {c} ,$ where $\mathbf {A}$ is the matrix whose elements are given by the $A_{ji}\!$ and $\mathbf {c}$ and $\mathbf {c} '$ are column vectors of the expansion coefficients of $|\psi \rangle$ and ${\hat {A}}|\psi \rangle ,$ respectively.

If we follow a similar line of reasoning for vectors in the dual vector space, we obtain $\mathbf {c} '^{\dagger }=\mathbf {c} ^{\dagger }\mathbf {A} ^{\dagger },$ where $\mathbf {A} ^{\dagger }$ is the transposed complex conjugate, or Hermitian adjoint, of $\mathbf {A} .$ We have thus justified our use of the term "Hermitian adjoint" in describing the action of an operator in the dual space.

Special Linear Operators in Quantum Mechanics

Hermitian Operator: An operator ${\hat {H}}$ is called Hermitian if ${\hat {H}}^{\dagger }={\hat {H}}.$ All physical observables in quantum mechanics are represented by Hermitian operators.

Anti-Hermitian operator: An operator ${\hat {A}}$ is called anti-Hermitian or skew-Hermitian if ${\hat {A}}^{\dagger }=-{\hat {A}}.$

Every operator ${\hat {C}}$ can be decomposed uniquely in terms of a Hermitian and an anti-Hermitian part: ${\hat {C}}={\frac {{\hat {C}}+{\hat {C}}^{\dagger }}{2}}+{\frac {{\hat {C}}-{\hat {C}}^{\dagger }}{2}}\equiv H+A$.

Unitary Operator: An operator ${\hat {U}}$ is called unitary if there exits an unique ${\hat {U}}^{-1}$ and is equal to ${\hat {U}}^{\dagger }$, i.e., ${\hat {U}}{\hat {U}}^{\dagger }={\hat {U}}^{\dagger }{\hat {U}}=1$. An important property of unitary operators is that they preserve the norm of a vector, which in quantum mechanics refers to the conservation of probability under physical operations. Most transformations of importance in quantum mechanics are given by unitary operators.

Antilinear operator: An operator ${\hat {A}}$ is called antilinear if for any two vectors $|\psi \rangle$ and $|\phi \rangle \in V$ and for any two complex numbers $c_{\psi }$ and $c_{\phi }$, ${\hat {A}}\left(c_{\psi }|\psi \rangle +c_{\phi }|\phi \rangle \right)=c_{\psi }^{\ast }{\hat {A}}|\psi \rangle +c_{\phi }^{\ast }{\hat {A}}|\phi \rangle .$ All operators of importance in quantum mechanics are linear, with one important exception - the time reversal operator ${\hat {T}}$ is an antilinear operator.

Theorem on Eigenvalues and Eigenstates of Hermitian Operator

We will now prove that the eigenvalues of Hermitian operators are real and that two eigenvectors of a Hermitian operator that correspond to different eigenvalues are orthogonal.

Proof: Consider an eigenstate $|\psi \rangle$ of a Hermitian operator ${\hat {H}}$, corresponding to an eigenvalue $\lambda ;$ i.e.,

${\hat {H}}|\psi \rangle =\lambda |\psi \rangle .$

Taking the Hermitian adjoint of both sides, and using the fact that $H^{\dagger }=H$, we get

$\langle \psi |{\hat {H}}=\lambda \langle \psi |.$

Taking the scalar product of the first equation with $\langle \psi |$ and of the second with $|\psi \rangle$, we get

$\langle \psi |{\hat {H}}|\psi \rangle =\lambda \langle \psi |\psi \rangle =\lambda ^{\ast }\langle \psi |\psi \rangle .$

Because $|\psi \rangle$ is not a null vector, we conclude that $\lambda ^{\ast }=\lambda ;$ i.e, $\lambda$ is real.

To prove the second part of the theorem, consider another eigenstate $|\phi \rangle$ with a different eigenvalue $\lambda '\neq \lambda ;$ i.e.,

${\hat {H}}|\phi \rangle =\lambda '|\phi \rangle .$

Taking the scalar product of the first equation with $\langle \phi |$ and of the second equation withn $\langle \psi |$, we get

$\langle \phi |{\hat {H}}|\psi \rangle =\lambda \langle \phi |\psi \rangle =\lambda '\langle \phi |\psi \rangle$.

Because $\lambda \neq \lambda ',$, we conclude that $\langle \phi |\psi \rangle =0;$ i.e., $|\psi \rangle$ and $|\phi \rangle$ are mutually orthogonal.

Since the eigenstates of a Hermitian operator are orthogonal (in fact, if they are normalized, then they are orthonormal), they often form a convenient basis in which to expand vectors; we will in fact often use the eigenstates of some observable as a basis.

Projection Operators

A projection operator is an operator that "projects" the vector that it acts on onto the direction of another given vector. The projection operator corresponding to the vector $|x\rangle$ is given by ${\hat {P}}=|x\rangle \langle x|.$ We may see that this is the case by acting with this operator on an arbitrary vector $|\psi \rangle :$

${\hat {P}}|\psi \rangle =|x\rangle \langle x|\psi \rangle =|x\rangle v_{x}$

We see that the result of the operation is a vector "parallel" to $|x\rangle ,$ as asserted.

Let us now state some properties of projection operators.

(1) A projection operator is Hermitian and equal to its own square; i.e., it is idempotent.

Proof: The fact that a projection operator is Hermitian follows immediately from the properties of the Hermitian adjoint stated above. The fact that it is also idempotent may be seen as follows:

${\hat {P}}^{2}=|x\rangle \langle x|x\rangle \langle x|=|x\rangle \langle x|={\hat {P}}$

(2) The product of two commuting projection operators ${\hat {P}}_{1}$ and ${\hat {P}}_{2}$ is also a projection operator.

Proof:

$({\hat {P}}_{1}{\hat {P}}_{2})^{\dagger }={\hat {P}}_{2}^{\dagger }{\hat {P}}_{1}^{\dagger }={\hat {P}}_{2}{\hat {P}}_{1}={\hat {P}}_{1}{\hat {P}}_{2}$

and

$({\hat {P}}_{1}{\hat {P}}_{2})^{2}={\hat {P}}_{1}{\hat {P}}_{2}{\hat {P}}_{1}{\hat {P}}_{2}={\hat {P}}_{1}^{2}{\hat {P}}_{2}^{2}={\hat {P}}_{1}{\hat {P}}_{2}.$

The sum of two projection operators is not, in general, a projection operator itself; it will only be a projection operator if the two original operators are orthogonal; i.e., if ${\hat {P}}_{1}{\hat {P}}_{2}=0.$