Axioms of Quantum Theory

Summary

We start with so called Hilbert space. It a vector space over the complex numbers $\mathbb{C}$. Vectors will be denoted $\ket{\psi}$ called as Dirac’s ket notation. It has a inner production $\braket{\psi\vert\phi}$ that maps an ordered pair of vectors to $\mathbb{C}$, defined by the properties

  1. Positivity: $\braket{\psi\vert\psi} \geq 0$ for $\ket{\psi} = 0$
  2. Linearity: $\bra{\phi}(a\ket{\psi_{1}} + b \ket{\psi_{2}}) = a \braket{\phi\vert\psi_{1}} + b \braket{\phi\vert\psi_{2}}$
  3. Skew symmetry: $\braket{\phi\vert\psi}^{*} = \braket{\psi\vert\phi}$.

It is complete in the norm $||\psi|| = \braket{\psi\vert\psi}^{1/2}$. An observable is a property of a physical system that in principle can be measured. In quantum mechanics, an observable is a self-adjoint operator i.e. $\mathbf{A} = \mathbf{A}^{\dagger}$.

An operator is a linear map taking vectors to vectors

$$ \begin{align*} \mathbf{A} : \ket{\psi} \to \mathbf{A}\ket{\psi} \\ \mathbf{A} (a\ket{\psi} + b\ket{\psi}) = a\mathbf{A}\ket{\psi} + b\mathbf{B}\ket{\psi} \end{align*} $$

The adjoint of the operator $\mathbf{A}$ is defined by

$$ \begin{align*} \braket{\phi\vert\mathbf{A}\psi} = \braket{\mathbf{A}^{\dagger}\phi\vert\psi} \end{align*} $$

for all vectors $\ket{\phi}, \ket{\psi}$ where we can write $\mathbf{A}\ket{\psi}$ as $\ket{\mathbf{A}\psi}$. Also note that, if $\mathbf{A}$ and $\mathbf{B}$ are self adjoint, then so is $\mathbf{A} + \mathbf{B}$ because $(\mathbf{A} + \mathbf{B})^{\dagger} = \mathbf{A}^{\dagger} + \mathbf{B}^{\dagger}$ but $(\mathbf{A}\mathbf{B})^{\dagger} = \mathbf{B}^{\dagger}\mathbf{A}^{\dagger}$. Also remember that $\mathbf{A}\mathbf{B} + \mathbf{B}\mathbf{A}$ and $i(\mathbf{A}\mathbf{B} + \mathbf{B}\mathbf{A})$ are always self-adjoint if $\mathbf{A}$ and $\mathbf{B}$ are.

A self-adjoint operator in a Hilbert space $\mathcal{H}$ has a spectral representation i.e. its eigen states form a complete orthonormal basis in $\mathcal{H}$. We can express a self-adjoint operator $\mathbf{A}$ as

$$ \begin{align*} \mathbf{A} = \sum_{n} a_{n}\mathbf{P}_{n} \end{align*} $$

Here each $a_{n}$ is an eigen value of $\mathbf{A}$ and $\mathbf{P}_{n}$ is the corresponding orthogonal projection onto the space of eigen vectors with eigen value $a_{n}$.

Note: If $a_{n}$ is non-degenerate (means it has same eigen values), then $\mathbf{P}_{n}=\ket{n}\bra{n}$; it is the projection onto the corresponding eigen vector.

Then $\mathbf{P}_{n}$’s satisfy

$$ \begin{align*} \mathbf{P}_{n}\mathbf{P}_{m} = \delta_{n,m}\mathbf{P}_{n}\\ \mathbf{P}_{n}^{\dagger} = \mathbf{P}_{n} \end{align*} $$

For unbounded operators in an infinite-dimensional space, the definition of self-adjoint and the statement of the spectral theorem are more subtle, but this need not concern us. Here, we don’t talk in-detail about unbounded operator so we refer you to look on functional analysis book.

The quantum measurement is done by the numerical outcome of a measurement of the observable $\mathbf{A}$ is an eigen value of $\mathbf{A}$ i.e. right after the measurement, the quantum state is an eigen state of $\mathbf{A}$ with the measured eigen value. If the quantum state just prior to the measurement is $\ket{\psi}$, then the outcome $a_{n}$ is obtained with probability

$$ \begin{align*} \mathbf{Prob} (a_{n}) = ||\mathbf{P}_{n}\ket{\psi}||^{2} = \braket{\psi\vert\mathbf{P}_{n}\vert\psi}. \end{align*} $$

If the outcome is $a_{n}$ is attained, then the *normalized* quantum state becomes $$ \begin{align*} \frac{\mathbf{P}_{n}\ket{\psi}}{\braket{\psi\vert\mathbf{P}_{n}\vert\psi}^{1/2}}. \end{align*} $$ Note that if the measurement is immediately repeated, then according to this rule, the same outcome is attained again, with probability one.

To study the dynamics, the time evolution of a quantum state is unitary; it is generated by a self-adjoint operator, called the Hamiltonian of the system. In the Schrödinger’s picture of dynamics, the vector describing the system moves in time as governed by the Schrödinger equation $$ \begin{align*} \frac{d}{dt}\ket{\psi(t)} = -i \mathbf{H}\ket{\psi(t)}, \end{align*} $$ where $\mathbf{H}$ is the Hamiltonian. We may reexpress this equation to first order in the infinitesimal quantity $dt$ as $$ \begin{align*} \ket{\psi(t + dt)} = (1 - i\mathbf{H}dt)\ket{\psi(t)}. \end{align*} $$ We can prove that $\mathbf{U}(dt) \equiv (1 - i\mathbf{H}dt)$ is unitary because $\mathbf{H}$ is self-adjoint and unitary operator satisfies $\mathbf{U}^{\dagger}\mathbf{U} = 1$ to linear order in $dt$. Since a product of unitary operators is finite, time evolution over a finite interval is also unitary i.e. $$ \begin{align*} \ket{\psi(t)} = \mathbf{U}(t)\ket{\psi(0)}. \end{align*} $$ In the case where $\mathbf{H}$ is time independent, we may write $\mathbf{U} = e^{-it\mathbf{H}}$. Note that, the Schrödinger equation is linear , while we are accustomed to non-linear dynamical equations in classical physics.

Hence, we have two distinct ways (called as dualism) for a quantum state to change. On the one hand, there is unitary evolution, which is deterministic. If we specify $\ket{\psi(0)}$, the theory predicts the state $\ket{\psi(t)}$ at a later time. But on the other hand, there is measurement which is probabilistic. The theory does not make definite predictions about the measurement outcomes; it only assigns probabilities to the various alternatives. This is troubling because it is unclear why the measurement process should be governed by different physical laws than other process.

Permalink at https://www.physicslog.com/physics-notes/qi-axioms-of-quantum-theory

Published on Jul 8, 2021

Last revised on Feb 2, 2023