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topic:qm:postulates
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| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| topic:qm:postulates [2022/12/09 18:27] – samuel | topic:qm:postulates [2023/02/15 18:26] (current) – samuel | ||
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| ====== Postulates of Quantum Mechanics ====== | ====== Postulates of Quantum Mechanics ====== | ||
| - | This follows the Postulates as written | + | //This follows |
| + | |||
| + | 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. | **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. | 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: |
| That is, the state $\ket{\psi}$ of the system at time t1 is related to the state $\ket{\psi' | That is, the state $\ket{\psi}$ of the system at time t1 is related to the state $\ket{\psi' | ||
| Line 16: | Line 19: | ||
| H is a fixed Hermitian operator known as the // | H is a fixed Hermitian operator known as the // | ||
| - | **3.** Quantum measurements are described by a collection ${M_m}$ of measurement operators. | + | **3.** |
| These are operators acting on the state space of the system being measured. | 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. | + | 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 | If the state of the quantum system is $\ket{\psi}$ immediately before the measurement then the probability that result $m$ occurs is given by | ||
| $$p(m) = \Braket{\psi | M_m^\dag M_m | \psi}$$ | $$p(m) = \Braket{\psi | M_m^\dag M_m | \psi}$$ | ||
| and the state of the system after the measurement is | and the state of the system after the measurement is | ||
| $$\frac{M_m \ket{\psi}}{\sqrt{\Braket{\psi | M_m^\dag M_m | \psi}}}.$$ | $$\frac{M_m \ket{\psi}}{\sqrt{\Braket{\psi | M_m^\dag M_m | \psi}}}.$$ | ||
| + | The measurement operators satisfy the // | ||
| + | $$\sum_m{M_m^\dag M_m} = I$$ | ||
| + | |||
| + | **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}, | ||
| + | |||
topic/qm/postulates.1670610445.txt.gz · Last modified: by samuel
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