Axioms of Quantum Theory


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.

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Published on Jul 8, 2021

Last revised on Feb 2, 2023