05. The Mathematical Structure of Quantum Theory

5.1 States

Complex Inner Product

Let $\psi ,\varphi$ be normalisable wave functions

$$⟨\varphi \mid \psi ⟩\equiv {\int }_{-\infty }^{\infty }\overline{\varphi (x)}\psi (x)dx\in \mathbb{C}$$

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Schwarz Inequality for Integrals

$${\left|{\int }_a^b\overline{\varphi (x)}\psi (x)dx\right|}^2\le {\int }_a^b|\varphi (x){|}^{2}dx{\int }_a^b|\psi (x){|}^{2}dx$$

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Square Norm

Let $\varphi$ be a normalisable wave function then

$$||\psi |{|}^2\equiv ⟨\psi \mid \psi ⟩$$

with norm $||\psi ||=\sqrt{||\psi |{|}^2}$

Note that $\varphi$ is normalised if $||\psi ||=1$

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Properties of Complex Inner Product Space

1. $\overline{⟨\varphi \mid \psi ⟩}=⟨\psi \mid \varphi ⟩$

2. $⟨\varphi \mid {\alpha }_1{\psi }_1+{\alpha }_2{\psi }_2⟩={\alpha }_1⟨\varphi \mid {\psi }_1⟩+{\alpha }_2⟨\varphi \mid {\psi }_2⟩$ for all ${\alpha }_1,{\alpha }_2\in \mathbb{C}$

3. $⟨{\beta }_1{\varphi }_1+{\beta }_2{\varphi }_2\mid \psi ⟩=\overline{{\beta }_1}⟨{\varphi }_1\mid \psi ⟩+\overline{{\beta }_2}⟨{\varphi }_2\mid \psi ⟩$ for all ${\beta }_1,{\beta }_2\in \mathbb{C}$

4. $||\psi |{|}^2\ge 0$

5. $||\psi |{|}^2=0$ if and only if $\psi =0$

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Complex Inner Product Space

Let $\mathcal{H}$ be a complex vector space with

1. Complex Inner Product $⟨\varphi ,\psi ⟩\in \mathbb{C}$ between vectors $\varphi ,\psi \in \mathcal{H}$

2. Satisfying the Properties of a Complex Inner Product Space

Then $\mathcal{H}$ is a complex inner product space

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States as Rays of $\mathcal{H}$

States of a Quantum System are elements of a Complex Inner Product Space $\mathcal{H}$

with proportional vectors representing the same state

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Hilbert Space

Hilbert Space is a Complete Complex Inner Product Space

where the states of a Quantum System are elements of the Hilbert Space

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5.2 Observables

Linear Operator on $\mathcal{H}$

Linear Operator is a pap $A:\mathcal{H}\to \mathcal{H}$ on complex vector space $\mathcal{H}$ satisfying

$$A({\alpha }_1{\psi }_1+{\alpha }_2{\psi }_2)={\alpha }_{1}A{\psi }_1+{\alpha }_{2}A{\psi }_2$$

for all ${\alpha }_i\in \mathbb{C}$ and ${\psi }_i\in \mathcal{H}$ for $i=1,2$

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Adjoint of a Linear Operator

Let $\mathcal{H}$ be a complex inner product space
Let $A$ be a Linear Operator on $\mathcal{H}$ then

Adjoint $A^{\ast }$ of $A$ satisfies (by definition)

$$⟨A^{\ast }\varphi \mid \psi ⟩=⟨\varphi \mid A\psi ⟩$$

for all $\varphi ,\psi \in \mathcal{H}$

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Self-Adjoint Linear Operator

Let $\mathcal{H}$ be a complex inner product space
Let $A$ be a Linear Operator on $\mathcal{H}$ then

If $A^{\ast }=A$ then

$$A\text{ is Self-Adjoint }$$

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Observables

Observables are self-adjoint linear operators $A$ on space of states $\mathcal{H}$

Note that observable refers to something measurable

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Position Operator

Position Operator $\mathbf{X}$ in three dimensions has components $X_i$ for $i=1,2,3$

Defined on wave functions $\psi \in \mathcal{H}$ by

$$(X_i\psi )(\mathbf{x})=x_i\psi (\mathbf{x})$$

where operator $X_i$ maps function $\psi$ to function $X_i\psi$

Note that the $i$th component of position operator $X_i$ multiplies wave function by $x_i$

Vector Notation

$$X\psi =\mathbf{x}\psi$$

One Dimension

$$X\psi =x\psi$$

with

$$||X\psi |{|}^2={\int }_{-\infty }^{\infty }{x}^2|\psi (x){|}^{2}dx$$

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Momentum Operator

Momentum Operator $\mathbf{P}$ has components $P_i$ for $i=1,2,3$

Defined on differentiable wave functions $\psi$ by

$$(P_i\psi )(\mathbf{x})=-i\hslash \frac{\partial \psi }{\partial x_i}(\mathbf{x})$$

Vector Notation

$$\mathbf{P}\psi =-i\hslash \nabla \psi$$

One Dimension

$$P\psi =-i\hslash {\psi }^{\prime }$$

where ${\psi }^{\prime }\equiv \frac{d\psi }{dx}$ with

$$||P\psi |{|}^2={\hslash }^2{\int }_{-\infty }^{\infty }|{\psi }^{\prime }(x){|}^{2}dx$$

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Properties of Self-Adjoint Linear Operators

Let $A,B$ be self-adjoint linear operators then

1. $(A+B)$ is self-adjoint where

$$(A+B)\psi (\mathbf{x})=A\psi (\mathbf{x})+B\psi (\mathbf{x})$$

2. $\alpha A$ is self-adjoint where $\alpha \in \mathbb{R}$ is a real constant

3. Composite Operator $AB$ is self-adjoint if $A$ and $B$ commute ($AB=BA$)

Proof

$(i)$ and $(ii)$ follow from definition almost immediately

For $(iii)$ then

$$⟨\varphi \mid AB\psi ⟩=⟨A\varphi \mid B\psi ⟩=⟨BA\varphi \mid \psi ⟩$$

Hence adjoint $(AB{)}^{\ast }=BA$ so by self-adjointness then $AB=BA$

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Hamiltonian Operator

Consider particle of mass $m$ moving in potential $V$ then

Hamiltonian operator is defined as

$$H\equiv \frac{|\mathbf{P}{|}^2}{2m}+V(\mathbf{X})$$

acting on differentiable wave functions by

$$(H\psi )(\mathbf{x})=-\frac{{\hslash }^2}{2m}{\nabla }^2\psi (\mathbf{x})+V(\mathbf{x})\psi (\mathbf{x})$$

One-Dimension

$$(H\psi )(x)=\left(\frac{P^2}{2m}+V(X)\right)\psi (x)=-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}(x)+V(x)\psi (x)$$

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Time-Dependent State Schrödinger Equation Operator Form

Schrödinger Equation for Time-Dependent States $\psi (t)$

$$i\hslash \frac{\partial }{\partial t}\psi (t)=H\psi (t)$$

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Stationary State Schrödinger Equation Operator Form

Consider Stationary State $\psi (t)=\psi e^{-iEt/\hslash }$ of energy $E$ then

$$H\psi =E\psi$$

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Non-Commutativity of Position and Momentum Operators

Consider $xp$ for motion in one-dimension then

$$\begin{array}{rl}(XP)\psi & =X(-ih{\psi }^{\prime })=i\hslash x{\psi }^{\prime }\\ (PX)\psi & =P(x\psi )=-i\hslash (x\psi {)}^{\prime }=-i\hslash x{\psi }^{\prime }-i\hslash \psi \end{array}$$

Hence

$$(XP-PX)\psi =ih\psi$$

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5.3 Measurement

Eigenstate

Let $A$ be an observable (self-adjoint linear operator on $\mathcal{H}$) then

Let state $\psi \in \mathcal{H}$ satisfy

$$A\psi =\alpha \psi$$

where $a\in \mathbb{C}$ is a constant

Then

$$\psi \text{ is an eigenstate of }A\text{ with eigenvalue }\alpha$$

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Spectrum

Spectrum is the set of all eigenvalues of $A$

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Realness of Eigenvalues of Observables

Let $\alpha$ be an eigenvalue of observable $A$ then

$$\alpha \text{ is real }$$

Proof

Consider $\alpha ⟨\psi \mid \psi ⟩$ then

$$\alpha ⟨\psi \mid \psi ⟩=⟨\psi \mid \alpha \psi ⟩=⟨\psi \mid A\psi ⟩=⟨A\psi \mid \psi ⟩=⟨\alpha \psi \mid \psi ⟩=\overline{\alpha }⟨\psi \mid \psi ⟩$$

As $⟨\psi \mid \psi ⟩\ne 0$ then $\alpha =\overline{\alpha }$ so $\alpha$ is real

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Orthogonality of Eigenstates of Distinct Eigenvalues

Suppose ${\psi }_1,{\psi }_2$ are eigenstates with distinct eigenvalues ${\alpha }_1\ne {\alpha }_2$

Then

$$⟨{\psi }_2\mid {\psi }_1⟩=0\text{ if }{\alpha }_1\ne {\alpha }_2$$

So ${\psi }_1$ and ${\psi }_2$ are orthogonal

Proof

As $A{\psi }_1={\alpha }_1{\psi }_1$ and $A{\psi }_2={\alpha }_2{\psi }_2$ then

Consider $⟨{\psi }_2\mid {\psi }_1⟩$

$$\begin{array}{rl}{\alpha }_1⟨{\psi }_2\mid {\psi }_1⟩& =⟨{\psi }_2\mid {\alpha }_1{\psi }_1⟩\\ & =⟨{\psi }_2\mid A{\psi }_1⟩\\ & =⟨A{\psi }_2\mid {\psi }_1⟩\\ & =⟨{\alpha }_2{\psi }_2\mid {\psi }_1⟩\\ & =\overline{{\alpha }_2}⟨{\psi }_2\mid {\psi }_1⟩\\ & ={\alpha }_2⟨{\psi }_2\mid {\psi }_1⟩\end{array}$$

As ${\alpha }_1\ne {\alpha }_2$ then $⟨{\psi }_2\mid {\psi }_2⟩=0$

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02 - Spectral Theorem

Spectral Theorem

Let $A:\mathcal{H}\to \mathcal{H}$ be a map where $\mathcal{H}$ is a finite-dimensional complex inner product space

If $A$ is self-adjoint then
There exists an orthonormal basis of eigenvectors for $A$

Note that for eigenvalues with two or more linearly independent eigenstates then use Gram-Schmidt

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Complete Set of Eigenstates for Observables axiom

Observable $A$ is required to have a complete set of eigenstates so
There exists a set of orthonormal eigenstates $\{{\psi }_n\}$ of $A$ where

$$A{\psi }_n={\alpha }_n{\psi }_n\text{ with }{a}_n\in \mathbb{R}\text{ and }⟨{\psi }_m\mid {\psi }_n⟩={\delta }_{mn}$$

Hence for any $\psi \in \mathcal{H}$ then it can be written as

$$\psi =\sum _n{c}_n{\psi }_n$$

where $c_n\in \mathbb{C}$ and $c_n=⟨{\psi }_n\mid \psi ⟩$

Note that if $\psi =\psi (t)$ then $c_n=c_n(t)$ (dependence on time)

Degenerate Form $\{a_n\}$ be the distinct eigenvalues where $\{{\psi }_{n,i}\}$ form an orthonormal basis for eigenspace with eigenvalue ${\alpha }_n$

Let

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Degeneracy of an Eigenvalue

Consider Eigenspace of a Eigenvalue then

Dimension $d$ of Eigenspace is the Degeneracy of Eigenvalue

Note

$$\psi =\sum _n\sum _{i=1}^{d_n}{c}_{n,i}{\psi }_{n,1}$$

where $d_n$ is degeneracy of eigenvalue ${\alpha }_n$ and $c_{n,i}\in \mathbb{C}$ and
${\psi }_{n,i}$ form a orthonormal basis for eigenspace with eigenvalue ${\alpha }_n$

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Non-Degenerate Eigenvalue

Eigenvalue $\alpha$ is non-degenerate if degeneracy $d=1$

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Quantum Measurement Postulate

Possible Measurement Outcomes of an observable $A$ are the eigenvalues $\{{\alpha }_n\}$ of $A$

Suppose system is in a normalised quantum state $\psi$ in form

If eigenvalue ${\alpha }_n$ is non-degenerate then probability of obtaining ${\alpha }_n$ in a measurement is

$${\mathbb{P}}_{\psi }(\text{obtaining }{\alpha }_n)=|c_n{|}^2=|⟨{\psi }_n\mid \psi ⟩{|}^2$$

If eigenvalue ${\alpha }_n$ is degenerate $d_n>1$ then

$${\mathbb{P}}_{\psi }(\text{obtaining }{\alpha }_n)=\sum _{i=1}^{d_n}|c_{n,i}{|}^2=\sum _{i=1}^{d_n}|⟨{\psi }_{n,i}\mid \psi ⟩{|}^2$$

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Orthogonality of the Hermite Polynomials

$${\int }_{-\infty }^{\infty }{H}_n(\xi )H_m(\xi )e^{-{\xi }^2}d\xi =0\text{ for }m\ne n$$

Any normalisable function $\psi$ on $\mathbb{R}$ can be written in an expansion in eigenstates of $H$

$$\psi (\xi )=\sum _{n=0}^{\infty }{c}_n{H}_n(\xi )e^{-{\xi }^2/2}$$

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5.4 Collapse of Wave Function

Collapse of Wave Function

Let $A$ be an observable

Upon measuring observable $A$ and getting non-degenerate eigenvalue ${\alpha }_n$ then

Immediately after measuring quantum state of system is eigenstate ${\psi }_n\in \mathcal{H}$ with

$$A{\psi }_n={\alpha }_n{\psi }_n$$

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Collapse of Wave Function - Degenerate Case

Let $A$ be an observable

If spectrum of observable $A$ has degenerate eigenvalues then
Quantum State $\psi$ as expansion from degeneracy of an eigenvalue
Immediately after measuring eigenvalue ${\alpha }_n$ then quantum state of system is

$$\psi \to \sum _{i=1}^{d_n}{c}_{n,i}{\psi }_{n,i}$$

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5.5 Summary

Deterministic and Probabilistic Process of Time Evolution

(Deterministic)
Normalised quantum state of system $\psi (t)$ evolves in time according to Schrödinger Equation
Given any observable $A$ then

$$\psi (t)=\sum _n{c}_n(t){\psi }_n$$

where $\{{\psi }_n\}$ are the complete orthonormal eigenstates of $A$
Typically chosen that $A=H$ so ${\psi }_n$ are stationary states in $E_n$ and

$$c_n(t)=c_n{e}^{-i{E}_{n}t/\hslash }\text{ where }{c}_n\in \mathbb{C}\text{ is constant in time}$$

(Probabilistic)
Suppose at time $t=t_0$ then observable $A$ is measured then

$${\mathbb{P}}_{\psi }(\text{obtaining eigenvalue }{\alpha }_n\text{ of }A)=|⟨{\psi }_n\mid \psi (t_0)⟩{|}^2$$

where ${\psi }_n$ is the eigenstate of $A$ with eigenvalue ${\alpha }_n$

By Collapse of Wave Function then immediately after obtaining value ${\alpha }_n$ then
Wave function collapses to $\psi (t_0)\to {\psi }_n$
Now start time evolution again using Schrödinger Equation with initial condition

$$\psi (t_0)={\psi }_n$$

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