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Loren Booda
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Does the history of wave packets translate exactly onto infinite phase space, or is phase space incompletely (or redundantly) covered by quantum mechanics?
So exists there a map from the coherent states (special quantum states) into phase space ?
Answer: more or less. To each point in phase space (classical state of motion) corresponds a coherent state that is "the best possible quantum description of the classical state". But the hilbert space is much bigger. There exist a lot of quantum states that do not have a classical counterpart.
Loren Booda said:vanesch,
Thanks for your tolerating my incomplete knowledge of quantum mechanics. You translated my question admirably. How would you define a special quantum state (a wave packet or coherent state) in relation to Hilbert space? Are there a finitude or infinity of quantum states (special or otherwise) compared to classical ones?
Phase space is a mathematical concept used to describe the state of a physical system. It is a multi-dimensional space where each point represents a specific combination of position and momentum for all particles in the system. In quantum dynamics, phase space is used to describe the behavior and evolution of quantum states over time.
In Hamiltonian mechanics, phase space is used to represent the state of a classical system. It consists of all possible values of position and momentum for all particles in the system, and is used to determine the equations of motion for the system. In quantum dynamics, the concept of phase space is extended to include the probabilistic nature of quantum states.
A one-to-one mapping in phase space means that each point in the space corresponds to a unique quantum state. This is important because it allows us to track the evolution of a quantum system over time by simply following the motion of a point in phase space.
No, phase space does not allow for the prediction of future behavior in quantum systems. This is because, in quantum mechanics, the state of a system cannot be known with certainty and is described by a wave function. Phase space simply represents all possible states and their probabilities, but does not provide specific predictions for the future behavior of a system.
In quantum statistical mechanics, phase space is used to describe the distribution of particles in a system. This is important for understanding the thermodynamic properties of a system, such as temperature and pressure. Phase space allows for the calculation of the probabilities of different states and their corresponding energies, which can then be used to calculate the thermodynamic properties of the system.