Field of Complex numbers

• The sets of rational (\(\mathbb{Q}\)), real (\(\mathbb{R}\)), and complex numbers (\(\mathbb{C}\)) are fields:
  • 2 operations: addition and multiplication
  • identity elements: addition (0), multiplication (1)
  • Inverse elements: addition (-x), multiplication (x^{-1})
  • Commutativity, associativity, distributivity

Complex numbers \(z \in \mathbb{C}\):

• Imaginary identity \(\quad i = \sqrt{-1}, \qquad i^2 = -1\)
• Complex conjugate \(z^*\): \(\quad z = x + i\, y \longrightarrow z^* = x - i \, y\)

Field of Complex numbers: properties

Assume \(\,\,z = x+iy \in \mathbb{C}\):

\[ \begin{aligned} \textrm{Representation} \qquad & z = x + i \, y = r e^{i \theta} = r \left(\cos\theta + i\,\sin\theta\right)\\ \textrm{Complex conjugate} \qquad & z^* = x - i \, y = r e^{-i \theta} = r \left(\cos\theta - i\,\sin\theta\right)\\ \textrm{Magnitude} \qquad & |z|^2 = z^* \, z = x^2 + y^2 = \Re\{z\}^2 + \Im\{z\}^2\\ \textrm{Phase} \qquad & \theta = -i\ln(z/|z|) = \arctan(y/x)\\ \textrm{Trigoniometry} \qquad & \cos\theta = \frac{e^{i\theta} + e^{-i \theta}}{2}, \qquad \sin\theta = \frac{e^{i\theta} - e^{-i \theta}}{2i}\\ \end{aligned} \]

Operations:

\[ \begin{aligned} \textrm{Addition} \qquad & z_1 + z_2 = (x_1 + x_2) + i\,(y_1 + y_2)\\ \textrm{Multiplication} \qquad & z_1 \, z_2 = r_1 r_2 e^{i (\theta_1 + \theta_2)}\\ \end{aligned} \]

vector spaces

A vector space \(\mathcal{V} = \left\{ | \alpha \rangle, | \beta \rangle, | \gamma \rangle, \dots\right\}\) over field \(F = \mathbb{C}\):

• Addition of vectors \(| \alpha \rangle + | \beta \rangle \in \mathcal{V}\)
• Scalar multiplication \(c | \alpha \rangle \in \mathcal{V}\)

Basis vectors

Linear independence

A vector \(| \xi \rangle\) is linearly independent of \(\left\{ |\alpha \rangle, |\beta \rangle, |\gamma \rangle, \dots \right\}\) 

\(\Leftrightarrow\) no linear combination: \(|\xi \rangle = a |\alpha \rangle + b |\beta \rangle + c |\gamma \rangle + \dots\)

Basis vectors

Suppose a finite set of \(n\) basis vectors:

\[ \left\{ |e_1\rangle, \, |e_2\rangle, \dots , \, |e_n\rangle \, \right\} \]

Each vector \(|\alpha \rangle\) can be written as superposition:

\[ |\alpha \rangle = a_1 |e_1\rangle + a_2 |e_2\rangle + \dots + a_n |e_n \rangle \]

In component notation for specific basis:

\[ |\alpha \rangle = ( a_1, a_2, \dots, a_n) \]

Normed vector space

• There exists a norm or length of a vector \(|\beta\rangle\) given by \(\| \beta \| \equiv \| \, |\beta \rangle \|\)

• Distance corresponding to norm:

\[ d(| \beta \rangle, | \alpha \rangle) = \| | \alpha \rangle - | \beta \rangle \| \]

• Example distance: \(\quad d\left( (x_1, y_1), \, (x_2, y_2)\right) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
• Example norm: \(\quad \| (3, 4) \| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\)

Inner product vector space

• An inner product of a vector space:

\[ \langle \, \langle \alpha | \, , \, | \beta \rangle \, \rangle = \langle \alpha | \beta \rangle \longrightarrow c \in \mathbb{C} \]

• The norm is defined by

\[ \| \beta \| = \sqrt{\langle \beta | \beta \rangle} \]

Orthonormal basis vectors

• A vector \(| \beta \rangle\) is normalized \(\quad \Leftrightarrow \quad \| \beta \| = 1\)
• A vector \(| \beta \rangle \perp | \alpha \rangle \quad \Leftrightarrow \quad \langle \alpha | \beta \rangle = 0\)
• Orthonormal set of vectors: \(\langle \alpha_i | \alpha_j \rangle = \delta_{ij}\)
• Always possible to find an orthonormal basis!

\(\longrightarrow\) In component notation: \(\langle \alpha | \beta \rangle = a_1^* b_1 + \dots + a_n^* b_n \qquad \textrm{with} \quad a_i = \langle e_i | \alpha \rangle\)

The norm is given by:

\[ \| \alpha \|^2 = \langle \alpha | \alpha \rangle = a_1^* b_1 + \dots + a_n^* b_n \qquad \textrm{with} \quad a_i = |a_1|^2 + \dots + |a_n|^2 \]

In \(\mathbb{R}^n\) the angle between two vectors is \({\bf a} \cdot {\bf b} = \|{\bf a}\| \|{\bf b}\| \cos(\theta)\):

\[ \cos \theta = \frac{\sqrt{\langle \alpha | \beta \rangle \, \langle \beta | \alpha \rangle}}{\| \alpha \| \| \beta \|} \]

Important theorems

• The dimension \(n\) (= number of basis vectors) is constant for a vector space.
• Gram-Schmidt procedure: any basis \(\longrightarrow\) orthonormal basis.
• Schwartz inequality:

\[ |\langle \alpha | \beta \rangle|^2 \leq \langle \alpha | \alpha \rangle \, \langle \beta | \beta \rangle \]

• Triangle inequality:

\[ \| \, |\alpha\rangle + |\beta\rangle \, \| \leq \| \alpha \|^2 + \| \beta \|^2 \]

Operators: linear transformations

• linear transformations \(\hat{T}\):

\[ |\alpha'\rangle = \hat{T} \, |\alpha\rangle \qquad \textrm{linearity: }\quad \hat{T}(c|\alpha\rangle + d|\beta\rangle) = a\,\hat{T}|\alpha\rangle + b\, \hat{T}|\beta\rangle \]

• If we know the basis vectors \(|e_1\rangle, \dots, |e_n\rangle\):

\[ \begin{aligned} |\alpha'\rangle & = \hat{T} \, |\alpha\rangle\\ & = \hat{T} \, \left( a_1|e_1\rangle + \dots + a_n|e_n\rangle \right)\\ & = \hat{T} \, a_1|e_1\rangle + \dots + \hat{T} \, a_n|e_n\rangle\\ & = a_1 \, \hat{T} \, |e_1\rangle + \dots + a_n \, \hat{T} \, |e_n\rangle\\ & = \sum_{i=1}^n a_i \, \hat{T} \, |e_i\rangle\\ \end{aligned} \]

Operators: matrix notation

• If we know the basis vectors \(|e_1\rangle, \dots, |e_n\rangle\):

\[ \hat{T} \, |\alpha\rangle = \sum_{j=1}^n a_j \, \hat{T} \, |e_j\rangle\\ \]

The \(\hat{T} \, |e_i\rangle\) can be written as superposition:

\[ \begin{aligned} \hat{T} \, |e_1\rangle & = T_{11}|e_1\rangle + T_{21}|e_2\rangle + \dots + T_{n1}|e_n\rangle \\ \hat{T} \, |e_2\rangle & = T_{12}|e_1\rangle + T_{22}|e_2\rangle + \dots + T_{n2}|e_n\rangle \\ & \dots \\ \hat{T} \, |e_n\rangle & = T_{1n}|e_1\rangle + T_{2n}|e_2\rangle + \dots + T_{nn}|e_n\rangle \\ \end{aligned} \]

\[ \Rightarrow \hat{T} \, |\alpha\rangle = \sum_{j=1}^n a_j \, \hat{T} \, |e_j\rangle = \sum_{j=1}^{n}\sum_{i=1}^{n} a_j T_{ij} | e_i \rangle = \sum_{i=1}^{n} \left(\sum_{j=1}^{n} T_{ij} a_j\right) | e_i \rangle \\ \]

Operators: matrix notation

\[ \Rightarrow \hat{T} \, |\alpha\rangle = \sum_{j=1}^n a_j \, \hat{T} \, |e_j\rangle = \sum_{j=1}^{n}\sum_{i=1}^{n} a_j T_{ij} | e_i \rangle = \sum_{i=1}^{n} \left(\sum_{j=1}^{n} T_{ij} a_j\right) | e_i \rangle \\ \]

Operator \(\hat{T}\) as a matrix \(T_{ij}\) for basis \(\left\{|e_1\rangle\, ,\, \dots\, , \, |e_n\rangle\right\}\)

\[ a'_i = \sum_{j=1}^{n} T_{ij} a_j \]

And the matrix:

\[ T_{ij} = \begin{pmatrix} T_{11} & T_{12} & \cdots & T_{1n}\\ T_{21} & T_{22} & \cdots & T_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ T_{n1} & T_{n2} & \cdots & T_{nn}\\ \end{pmatrix}\qquad \textrm{with } \quad T_{ij} = \langle e_i | \hat{T} | e_j \rangle \]

Matrices and vectors

If we have a basis basis \(\left\{|e_1\rangle \, , \dots \, , \,| e_n \rangle \right\}\)

\[ |\alpha\rangle = \begin{pmatrix} a_1\\ a_2\\ \vdots\\ a_n\\ \end{pmatrix} \]

An operator acting on a vector \(|\alpha\rangle\): \[ \hat{T} |\alpha\rangle \longrightarrow \sum_{j=1}^n T_{ij} a_j = \begin{pmatrix} T_{11} & T_{12} & \cdots & T_{1n}\\ T_{21} & T_{22} & \cdots & T_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ T_{n1} & T_{n2} & \cdots & T_{nn}\\ \end{pmatrix}\, \begin{pmatrix} a_1\\ a_2\\ \vdots\\ a_n\\ \end{pmatrix} \]

Operators and matrix properties

• Adding two operators:

\[ \hat{U} = \hat{S} + \hat{T} \longrightarrow U_{ij} = S_{ij} + T_{ij} \]

• Performing multiple operators \(\hat{U} = \hat{S} \hat{T}\):

\[ \hat{U} |\alpha\rangle = \hat{S} \hat{T} |\alpha\rangle \longrightarrow U_{ij} = \sum_k S_{ik} T_{kj} \]

Intermezzo: Matrix products

The matrix product between matrices \(A\) and \(B\) is defined as

\[ \begin{aligned} A \cdot B & = \pmatrix{ a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ } \pmatrix{ b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23}\\ b_{31} & b_{32} & b_{33}\\ }\\ &\\ & = \sum_{j} a_{ij} b_{jk}\\ \end{aligned} \]

• Rows of \(A\) are multiplied by columns of \(B\).
• \(A_{MN} \cdot B_{NK} \leftarrow\) No. columns of \(A\) must equal No. rows of \(B\)

Operators and matrix properties

• Transpose of a matrix \(\quad \tilde{T} = T_{ji}\)
  • symmetric: \(\qquad\,\,\tilde{T} = T\)
  • antisymmetric: \(\quad\tilde{T} = -T\)
• Complex conjugate of a matrix \(\quad T^* = T_{ij}^*\)
  • real: \(\qquad\quad\,\, T^* = T\)
  • imaginary: \(\quad T^* = -T\)
• Hermitian conjugate of a square matrix \(\quad T^\dagger = \tilde{T}^* = T_{ji}^*\)
  • Hermitian: \(\qquad\quad T^\dagger = T \qquad\)
  • skew hermitian: \(\quad T^\dagger = -T\)

Bra-ket notation and inner products

• The inner product for orthonormal basis \(\left\{|e_1\rangle \, , \dots \, , \,| e_n \rangle \right\}\)

\[ \langle \alpha | \beta \rangle = a_1^* b_1 + a_2^* b_2 + \dots + a_n^* b_n = {\bf a}^\dagger {\bf b} \]

• ket \(| \beta \rangle\) is a column vector
• bra \(\langle \alpha |\) is a complex conjugate row vector

In vector notation: \[ \langle \alpha | \longrightarrow \vec{a} = \pmatrix{a^*_1 & a^*_2 & \dots & a^*_N} \qquad |\beta\rangle \longrightarrow \vec{b} = \pmatrix{b_1 \\ b_2 \\ \vdots \\ b_N} \]

Operators and matrix properties

• Transpose of a matrix product \(\tilde{S\,T} = \tilde{T}\tilde{S}\)
• Hermitian of a matrix product \(\left(S\,T\right)^\dagger = T^\dagger S^\dagger\)
• Inverse matrix \(T^{-1} T = T\, T^{-1} = \mathbb{1} = \delta_{ij}\)
• Inverse of a matrix product \(\left( S\, T\right)^{-1} = T^{-1} S^{-1}\)
• Unitary matrix \(U^\dagger = U^{-1}\)
• Unitary operators preserve inner product:

\[ \langle \alpha' | \beta' \rangle = {\bf a'}^\dagger {\bf b'} = (U{\bf a})^\dagger ({U \bf b}) = {\bf a}^\dagger U^\dagger U {\bf b} = {\bf a}^\dagger {\bf b} = \langle \alpha | \beta \rangle \]

Change of basis

• Unitary matrices \(\,U (\longleftarrow U^\dagger = U^{-1})\) preserve inner product
  • Norm doesn’t change
  • Angles between vectors don’t change

\(\longrightarrow\) Apply unitary transformation to orthonormal basis is again orthonormal basis

\[ \{ |e_1\rangle, |e_2\rangle, \dots , |e_n\rangle \} \qquad | e'_i \rangle = U | e_i \rangle \quad\textrm{ is orthonormal} \]

If \(T\) transforms a basis: \(a_i \rangle = T | e_i \rangle\) to another orthonormal one: \(\langle a_j | a_i \rangle = \delta_{ij}\) \(\Longrightarrow T\) is unitary:

\[ \begin{aligned} \delta_{ij} & = \langle a_j | a_i \rangle\\ & = \langle a_j | T | e_i \rangle\\ & = \langle e_j | T^\dagger T | e_i \rangle \\ \end{aligned} \quad\Rightarrow\quad T^\dagger T = \mathbb{1} \quad\Rightarrow\quad T^\dagger = T^{-1} \]

Commutators

• Matrix-multiplication not commutative \(\longleftrightarrow\) Order of operators!
• Commutator of two operators/matrices

\[ [\hat{S}, \hat{T}] = \hat{S}\hat{T} - \hat{T}\hat{S} \longleftrightarrow [S, T] = S\,T - T\,S \]

• Anti-commutator of two operators/matrices

\[ \{\hat{S}, \hat{T}\} = \hat{S}\hat{T} + \hat{T}\hat{S} \longleftrightarrow \{S, T\} = S\,T + T\,S \]

Eigenvalue problems

Eigenvector \({\bf x} \neq {\bf 0}\) and eigenvalues \(\lambda\) of matrix \(A\):

\[ A {\bf x} = \lambda {\bf x} \Leftrightarrow (\lambda \mathbb{1} - A) {\bf x} = {\bf 0}\\ \]

Because \({\bf x} \neq {\bf 0}\) the inverse of \(\lambda \mathbb{1} - A\) cannot exist, because if it would:

\[ \begin{aligned} (\lambda \mathbb{1} - A) {\bf x} & = {\bf 0}\\ \Longrightarrow (\lambda \mathbb{1} - A)^{-1}(\lambda \mathbb{1} - A) {\bf x} & = (\lambda \mathbb{1} - A)^{-1} {\bf 0}\\ \Longrightarrow (\lambda \mathbb{1} - A)^{-1}(\lambda \mathbb{1} - A) {\bf x} & = {\bf 0}\\ \Longrightarrow {\bf x} & = {\bf 0} \end{aligned} \]

Eigenvalue problems

• Matrix \((\lambda \mathbb{1} - A)\) not invertible \(\longrightarrow\) the determinant has to be zero
• Solve characteristic equation:

\[ \det(\lambda \mathbb{1} - A) = 0 \]

• Determinant is a “characteristic” polynomial in \(\lambda\)
• Highest order of \(\lambda\) is the dimension \(N\) of the \(N\times N\) matrix
• Solving it means finding \(\lambda\) values

Example eigenvalue problem

\[ A = \pmatrix{ -5 & 2\\ -7 & 4\\ } \]

This gives for the characteristic equation: \(\quad\det(\lambda \mathbb{1} − A) = 0\):

\[ \begin{aligned} \det\left[ \lambda \pmatrix{ 1 & 0\\ 0 & 1\\ } - \pmatrix{ -5 & 2\\ -7 & 4\\ } \right] = 0\\ \\ \Longrightarrow \det\left[ \pmatrix{ \lambda+5 & -2\\ 7 & \lambda-4\\ } \right] = 0\\ \end{aligned} \]

The determinant is:

\[ \lambda^2 + \lambda − 6 = 0 \longrightarrow (\lambda - 2)(\lambda + 3) = 0 \]

Example eigenvalue problem CTU’d

• Find eigenvalues \(\lambda_i\)
• Eigenvectors by filling in a specific eigenvalue \(\lambda_i\)

\[ A {\bf x} = \pmatrix{ -5 & 2\\ -7 & 4\\ } \qquad \lambda_1 = 2, \quad\lambda_2 = -3 \]

Eigenvector \(\,{\bf x}_1 = (x, y)\,\,\) for \(\,\,\lambda_1 = 2\)

\[ \begin{aligned} A = \pmatrix{ \lambda_1+5 & -2\\ 7 & \lambda_1-4\\ } \pmatrix{ x\\ y\\ } = \pmatrix{ 7 & -2\\ 7 & -2\\ } \pmatrix{ x\\ y\\ } = 0\\ \Longrightarrow {\bf x} = c \pmatrix{ 2\\ 7\\ }\\ \end{aligned} \]

Eigenvalue problems: large matrices

• Inverse exists \(\Leftrightarrow\) determinant is nonzero
• Determinants of \(3\times 3\) or higher order matrices \(A\): \[ \begin{aligned} \det(A) & = \det\pmatrix{ a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\} \\ & \\ & = \begin{vmatrix} a_{22} & a_{23}\\ a_{32} & a_{33}\\ \end{vmatrix} a_{11} - \begin{vmatrix} a_{21} & a_{23}\\ a_{31} & a_{33}\\ \end{vmatrix} a_{12} + \begin{vmatrix} a_{21} & a_{22}\\ a_{31} & a_{32}\\ \end{vmatrix} a_{13} \\ &\\ & = (a_{22} a_{33} - a_{23} a_{32}) a_{11} - \dots \\ \end{aligned} \]

Characteristic polynomial in \(\lambda\) of order \(N\) for \(N\times N\) matrix

Eigenvalue problems: simplify

• Reduce matrix \(A\) to simpler matrix \(B\)
• Transform matrix \(A\) by invertible matrix \(T\):

\[ B = T^{-1} A T \qquad \Longrightarrow \qquad \{\lambda_i\} \quad\textrm{the same} \]

• Characteristic equation of upper (or lower) triangle matrices \(B\):

\[ (\lambda - b_{11})(\lambda - b_{22}) \,\dots\, (\lambda - b_{nn}) = 0 \]

• Derive eigenvalues and eigenvectors for \(B\):

\[ \Longrightarrow \quad \left\{ \begin{aligned} &\textrm{Eigenvalues}\qquad \lambda_i = b_{ii}\\ &\textrm{Eigenvectors}\quad {\bf x'}_i\,\,\textrm{of}\,\,B = T {\bf x}_i\\ \end{aligned} \right. \]

Matrix-formalism of quantum mechanics

• Works if only a finite sum of basis functions is used
• Approximations possible ?

! General case is PROBLEMATIC !

• Often: infinite number of basis functions
• Inner products might not be finite \(\longrightarrow\) not normalizable
• Operators can have infinite expectation values ? Undefined ?

General Quantum mechanical formalism

Mathematical correspondence:

• States: vectors in Hilbert space: \(\qquad\qquad\qquad\quad L^2\) square integrable functions
• Observables: Hermitian operators: \(\qquad\qquad\qquad T^\dagger = T\)
• Measurements: Orthogonal projections
• Symmetries of the system: unitary operators: \(\qquad U^\dagger = U^{-1}\)

Dirac “bra-ket” notation: \(\qquad \langle \textrm{bra} |, \quad| \textrm{ket}\rangle\)

• A convenient way of writing
• Implicitly expresses the mathematical properties.

Pre-Hilbert spaces or Banach spaces

A Cauchy series:

• an (infinite) sequence of vectors \(v_n \in \mathcal{V}: v_1, v_2, v_3, \dots\)
• has property: for every small value \(\epsilon\) we can find a finite \(N\):

\[ \forall\,\, m,\, n > N: \quad \| v_n - v_m \| < \varepsilon \quad\textrm{ with } v_n, v_m \in \mathcal{V} \]

• A Cauchy series converges to a certain “vector” \(v\) that can be outside \(\mathcal{V}\).

A Banach space:

• Is a normed vector space
• Every Cauchy series converges to an element \(v\) of the vector space: \(v \in \mathcal{V}\).
  • Example: any Cauchy series of real numbers \(x_n \in \mathbb{R}\) converges in \(\mathbb{R}\)
  • Example: Cauchy series of rational numbers \(x_n = \frac{1}{2^n} \in \mathbb{Q}\) doesn’t converge in \(\mathbb{Q}\)

Hilbert spaces

A Hilbert space

• Has an inner product
• Has its norm derived from the inner product: \(\| \alpha \| = \sqrt{\langle \alpha | \alpha \rangle }\)
• Is a Banach space

Summary of vector spaces/properties

Wave functions in Hilbert space

Quantum mechanics \(\longrightarrow\) specific Hilbert space: \(L^2(a, b)\)

• functions \(f(x)\) square integrable over interval \([a, b]\)

\[ \|f \|^2 = \int_a^b |f(x)|^2 dx < \infty\\ \Longrightarrow f(x) \quad\textrm{normalizable} \]

Wave functions in Hilbert space

• Schwartz inequality \(\Longrightarrow\) inner product is finite

\[ |\langle f | g \rangle| \leq \sqrt{\langle f | f \rangle \langle g | g \rangle} \]

• Orthonormal complete set of basis vectors \(\{ |f_n\rangle\}\)

\[ \langle f_m| f_n \rangle = \int_a^b f_m(x)^*f_n(x) dx = \delta_{mn} \]

\[ | f \rangle = \sum_n c_n \, | f_n \rangle, \qquad c_n = \langle f_n| f \rangle = \int_a^b f_n(x)^* f(x) dx \]

\(\longrightarrow\) We will use sometimes \(f\), \(g\) instead of \(|\psi\rangle\), \(|\psi_n\rangle\), etc. for (wave) functions

Observables

• Observables are represented by measurement operators

\[ \langle Q \rangle = \int \Psi^* \hat{Q} \Psi \, dx = \langle \Psi | \hat{Q}\Psi\rangle \]

Since measurements need to be real: \(\langle Q \rangle = \langle Q \rangle^*\)

\[ \langle \Psi | \hat{Q}\Psi\rangle = \langle \hat{Q}\Psi | \Psi\rangle \]

\(\Longrightarrow\) The operator \(\hat{Q} = \hat{Q}^\dagger\) is Hermitian

• In a finite basis: \(\quad\) Hermitian operators \(\,\,\Longleftrightarrow\,\,\) Hermitian matrices

Which operators are Hermitian?

• Check this for \(\,\,\hat{p} = -i \hbar \frac{d}{dx}\):

\[ \begin{aligned} \langle f | \hat{p} g \rangle & = \langle f | -i\hbar \frac{d}{dx} g \rangle\\ & = - i\hbar \int f(x)^* \frac{d g(x)}{dx} dx\\ & = - f(x)^* g(x) \Big|^{+\infty}_{-\infty} + i\hbar \int \frac{d f(x)^*}{dx} \, g(x) dx\\ & = i\hbar \int \frac{d f(x)^*}{dx} \, g(x) dx\\ & = \langle -i\hbar \frac{d}{dx} f | g \rangle \\ & = \langle \hat{p} f | g \rangle \\ \end{aligned} \]

\(\longrightarrow\) Important that \(f\) and \(g\) become zero at \(x = \pm \infty\)

Determinate states of observables

• Perform independent measurements \(\longrightarrow\) different outcomes (probabilistic)
• A determinate state \(\longrightarrow\) every time the same outcome
• For a determinate state \(|\Psi\rangle\) for \(Q\): \(\quad Q \longrightarrow \langle Q \rangle = q\) is a constant

Spectrum: eigenvalues of an operator

• Spectrum of an operator: all eigenvalues
• Multiplicity or degeneracy: same eigenvalue for 2 or more eigenstates
• Hamiltonian operator is the standard example

\[ \hat{H} |\psi \rangle = E |\psi \rangle \]

• Two types of spectra:
  • Discrete spectrum: spaced eigenvalues, normalizable eigenstates (e.g. infinite well)
  • Continuous spectrum: Continuous range of eigenvalues, non-normalizable eigenstates (e.g. free particle)
  • Possible mixture of both (e.g. finite well)

Spectrum: eigenvalues of an operator

Discrete spectrum

1. Eigenvalues of operator \(\hat{Q}\) are real:

\[ \textrm{Assume eigenvalue } q \quad \hat{Q} f = q f \]

\[ \Longrightarrow q \langle f|f \rangle = \langle f|\hat{Q} f \rangle = \langle \hat{Q} f | f \rangle = q^* \langle f|f \rangle \]

Discrete spectrum

Properties

1. Real eigenvalues
2. Eigenfunction of different eigenvalues are orthogonal: \(\quad \langle f_m | f_n \rangle = \delta_{mn}\)
3. Degenerate eigenvalues can exist, but we can choose orthonormal basis of those eigenfunctions
4. Finite dimensional spaces are complete

Axiom: Any observable operator in Hilbert space has a complete basis of eigenfunctions

Discrete spectrum: statistical interpretation

• wave function \(\Psi(x, t)\) and eigenfunctions \(f_n: \quad \hat{Q} f_n = q_n f_n\)
• Wave function can be expanded in \(f_n\):

\[ \Psi(x, t) = \sum_n c_n(t) f_n(x), \qquad \textrm{with}\quad c_n(t) = \langle f_n | \Psi \rangle = \int f_n(x)^* \Psi(x, t) dx \]

• Measure expectation with observable operator \(\hat{Q}: \quad \langle \Psi | \hat{Q} \, \Psi\rangle\)

\[ \begin{aligned} \langle \hat{Q} \rangle = \langle \Psi | \hat{Q} \, \Psi \rangle & = {\LARGE\langle} \sum_m c_m(t) f_m(x) {\LARGE|} \hat{Q} \,\sum_n c_n(t) f_n(x) {\LARGE\rangle}\\ & = \sum_m\sum_n c_m(t)^* c_n(t) q_n \langle f_m(x) | f_n(x)\rangle\\ & = \sum_m\sum_n c_m(t)^* c_n(t) q_n \delta_{mn} = \sum_n |c_n(t)|^2 q_n\\ \end{aligned} \]

Spectrum: eigenvalues of an operator

Intermezzo: The Dirac Delta function

Continuous spectra

• Eigenfunctions/values continuous variable \(z \longrightarrow f_{z}\)
• Eigenfunctions are NOT normalizable
• Solution: Assume real eigenvalues
• New definitions:

\[ \textrm{Orthonormality} \qquad \langle f_{z'}| f_{z}\rangle = \delta(z' - z) \]

Continuous spectra: example

Momentum operator for a free particle

Eigenvalues and eigenfunctions:

\[ -i \hbar \frac{d}{dx} f_p(x) = p f_p(x) \quad \textrm{with} \qquad f_p(x) = A e^{ipx/\hbar} \]

If eigenvalues \(\,\,p\in \mathbb{R}\) then \(\{f_p\}\) is orthogonal:

\[ \langle f_{p'} | f_p\rangle = \int f_{p'}^* f_p dx = |A|^2 \int e^{i(p - p')x/\hbar} dx = |A|^2 2\pi \hbar \delta(p - p') \]

Continuous spectra: example

Momentum operator for a free particle

Completeness follows from Fourier analysis:

\[ f(x) = \int c(p) f_p(x) dp = \frac{1}{\sqrt{2\pi\hbar}} \int c(p) e^{ipx/\hbar} dp \]

The coefficients \(c(p)\) are as expected:

\[ \langle f_{p'} | f_p \rangle = \int c(p) f_{p'}^* \, f_p \, dp = \int c(p) \delta(p - p') dp = c(p') \]