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* First principle: superposition | * First principle: superposition | ||
− | ** with each physical system there is an associated Hilbert space <math>{\ | + | ** with each physical system there is an associated Hilbert space <math>{\rm E} _ {\rm H}</math> |
** at each time t, the state of the system is completely determined by a normalized vector <math>|\psi(t) \rangle</math> | ** at each time t, the state of the system is completely determined by a normalized vector <math>|\psi(t) \rangle</math> | ||
* Second principle: measurement | * Second principle: measurement | ||
− | ** with each physical quantity <math>A \ </math> one can associate a linear hermitian operator <math>\hat{A} \ </math> | + | ** with each physical quantity <math>A \ </math> one can associate a linear hermitian operator <math>\hat{A} \ </math> mapping <math>{\rm E} _ {\rm H}</math> into <math>{\rm E} _ {\rm H}</math>. <math>\hat{A} \ </math> is the observable representing <math>A \ </math>. |
+ | ** If <math>|\psi(t) \rangle</math> denotes the state of the system before a measurement, then the only possible results of a measurement are the eigenvalues <math>a_ \alpha \ </math> of <math>\hat{A} \ </math> | ||
+ | ** The probability of measuring <math>a_ \alpha \ </math> is <math>\| \psi _ \alpha \|</math> where |
Latest revision as of 16:39, 7 May 2006
major concepts
the three principles of quantum mechanics. page 101.
- First principle: superposition
- with each physical system there is an associated Hilbert space EH
- at each time t, the state of the system is completely determined by a normalized vector
- Second principle: measurement
- with each physical quantity one can associate a linear hermitian operator mapping EH into EH. is the observable representing .
- If denotes the state of the system before a measurement, then the only possible results of a measurement are the eigenvalues of
- The probability of measuring is where