1. The Wigner Function Chen Levi

2. Eugene Paul Wigner Received the Nobel Prize for Physics in 1963 1902 - 1995

3. The Phase Space q p Harmonic Oscillator Bohr quantization rule: q – position p – momentum Liouville’s theorem: Distribution function in phase space may change shape in time but do not change volume – conservation of probability

4. Distribution Functions Phase-space formulation of quantum mechanics allows us to work with constant-number equations instead of operators Let be arbitrary operator (observable)Expectation value of : - Distribution Function

5. Example: On the other hand if we will take:

6. Houston we have a problem: The problem arises because there is no unique way to assign quantum-mechanical operator to a classical function. Therefore we can not uniquely define the distribution function without specified the rule of association.

7. We can define a general class of the quantum phase-space distribution functions: Choosing defines the rule of association and therefore defines the distribution function In our example:

8. From the definition: We can get:

9. By choosing: We will get the Wigner Distribution function: In pure state and we get:

10. The Wigner Distribution Function Given a wavepacket Fourier transform of A way to present probability density of a quantum system in the phase space

12. Harmonic Oscillator lowest eigenstate Example: Wigner transform of a Gaussian wavepacket is a Gaussian phase space distribution q p

13. Harmonic Oscillator sixth eigenstate p q Note that the Wigner distribution function can take on negative values. However, these values never survive in the calculation of an observable becauseand are always positive!!!

14. p q Gaussian wavepacket in a harmonic oscillator potential Coherent state: The motion of the center of the Wigner distribution is entirely classical

15. p q Squeezed state: p q