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Dirac’s Quantum Condition

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1 Dirac’s Quantum Condition
Classical mechanics relates two conjugated variables by using the Poisson bracket. Dirac’s quantum condition extends this relation to quantum mechanical operators: The commutator between two operators must relate to the classical Poisson bracket between the two corresponding functions through the following relationship HW #2 Operators which follow Dirac’s quantum condition form an internally consistent set, though there may be more than one set (different representations)

2 Poisson bracket and commutator
For example, for f(x,p)= x and g(x,p)=p the Poisson bracket is

3 Variance Finally, we can connect everything we know about commutators and the Dirac’s quantum condition and obtain the most fundamental property of the Quantum World individual measurements expectation value

4 Uncertainty Principle
For the product of the standard deviations of two properties of a quantum mechanical system whose state wavefunction is y, it can be shown (you’ll do it in your HW2)

5 The Born Interpretation
The explicit function representing a state of a system in a particular coordinate system and in a particular representation is called WAVEFUNCTION:  Remember? Probability of an event is given by |f|2 where f is a complex number (probability amplitude)

6 cont Normalization: The probability of finding the particle SOMEWHERE in space and time MUST be =1 Wavefunctions which obey the eigenvalue equation are called eigenfunctions

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