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--- topic:qm:postulates [2022/12/09 16:26] – samuel
+++ topic:qm:postulates [2023/02/15 18:26] (current) – samuel
@@ Line 1: / Line 1: @@
 ====== Postulates of Quantum Mechanics ======
-This follows the Postulates as stipulated by \cite{Nielsen.Chuang:2010}.
+//This follows (verbatim) the Postulates as written in \cite{Nielsen.Chuang:2010}.//
+Four postulates are defined to outline the space quantum mechanics exists in. 1., describes the state of a closed system. 2. describes the evolution of such system in time. 3. describes the influence of observations or information gathering from such a system and 4. defines how multiple systems can be combined.
 **1.** Associated to any isolated physical system is a complex vector space with inner product (that is, a Hilbert space) known as the state space of the system.
 The system is completely described by its state vector, which is a unit vector in the system’s state space.
-**2.** The evolution of a closed quantum system is described by a unitary transformation.
+**2.** The evolution of a closed quantum system is described by a [[topic:math:linalg:operator|unitary]] transformation.
 That is, the state $\ket{\psi}$ of the system at time t1 is related to the state $\ket{\psi'}$ of the system at time t2 by a unitary operator U which depends only on the times t1 and t2, $\ket{\psi'} = U \ket{\psi}$
+**2.'** The time evolution of the state of a closed quantum system is
+described by the //Schrödinger equation//,
+$$ i\hbar \partial_t \ket(\psi) = H \ket(\psi).$$
+In this equation, $\hbar$ is a physical constant known as //Planck’s constant// whose value
+must be experimentally determined. The exact value is not important to us. In
+practice, it is common to absorb the factor $\hbar$ into $H$, effectively setting $\hbar = 1$.
+H is a fixed Hermitian operator known as the //Hamiltonian// of the closed system.
+**3.** [[topic:qm:measurements|Quantum measurements]] are described by a collection ${M_m}$ of measurement operators.
+These are operators acting on the state space of the system being measured.
+The index $m$ refers to the measurement outcomes that may occur in the experiment.
+If the state of the quantum system is $\ket{\psi}$ immediately before the measurement then the probability that result $m$ occurs is given by
+$$p(m) = \Braket{\psi | M_m^\dag M_m | \psi}$$
+and the state of the system after the measurement is
+$$\frac{M_m \ket{\psi}}{\sqrt{\Braket{\psi | M_m^\dag M_m | \psi}}}.$$
+The measurement operators satisfy the //completeness// equation ((This comes from all probabilities $p(m)$ for the outcomes have to give $1$.)),
+$$\sum_m{M_m^\dag M_m} = I$$
+**4.** The state space of a composite physical system is the tensor product of the state spaces of the component physical systems.
+Moreover, if we have systems numbered $1$ through $n$, and system number $i$ is prepared in the state
+\ket{\psi_i}, then the joint state of the total system is $\ket{\psi_i} \otimes \ket{\psi_i} \otimes \ldots \otimes \ket{\psi_i}$.

Kollektanee - Samuel Gyger

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