Understanding Probability Amplitude, State Operators and Galilei Group

In summary: As the universe cooled, the energy became spread out and the universe entered a lower-energy state. In summary, the wavefunction is the dimensionless complex number that represents the probability amplitude of what will happen in a given situation. The value of the wavefunction is 'probability amplitude' in discrete case and 'probability amplitude density' in continuous case. The former is a dimensionless complex number and the latter is the same multiplied by the inverse of the units of coordinare vectors in space on which the distribution is defined. Eg. Ψ(x,t) would have units ##L^{-3}##. The actual
  • #1
Delta Kilo
329
22
Greetings,

Just checking if I'm getting this ... please correct me if I'm wrong.

The value of the wavefunction is 'probability amplitude' in discrete case and 'probability amplitude density' in continuous case. The former is a dimensionless complex number and the latter is the same multiplied by the inverse of the units of coordinare vectors in space on which the distribution is defined. Eg. Ψ(x,t) would have units ##L^{-3}##. The actual scale factor is irrelevant.

State operator ρ is dimensionless with Tr{ρ}=1. However for the operator R representing observable the eigenvalues must have proper units if we want the measurements to come out right. That is,λ in ##R|ψ\rangle=λ|ψ\rangle## has to have the right units. Now what does it mean in terms of domain and range of operator R? I thought it was supposed to be defined over some vector space ##\Omega^\times##, but how can ##| ψ\rangle## and ##(5kg)|ψ\rangle## belong to the same space? I guess these are in fact two different spaces with the same structrure but as you never need to add elements from different spaces and the units always cancel out nicely in the end, the distinction is quietly ignored.

Looking at the generators of Galilei group, the units of P are necesserily ##L^{-1}##, H is ##T^{-1}##, J is 1 and G is ##L^{-1}T## (so we can have eg. ##e^{iPx}##, ##e^{iHt}## etc. But in order to connect them to familiar notions, we introduce this funny coefficient ##\hbar## which has units of action ##J \cdot s##. But I 'm just wondering, is there some deeper meaning whereby measuring energy in ##s^{-1}## and momentum in ##m^{-1}## would actually make sense? Let's say we introduce a unit of action , ##\hbar=1 fubar##.Then we'll be measuring energy in fubars per second and momentum in fubars per meter. Mass will come out as fubar-second/meter² (hmm, not sure about this one). What would the quantity of 1 fubar signify?

Regards, DK
 
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  • #2
Planck's constant has units of angular momentum or "action". In old quantum physics, Bohr-Sommerfeld quantization was quantization of action.
 
  • #3
The best way of looking at Planks constant IMHO is the path integral formalism:
http://hitoshi.berkeley.edu/221a/pathintegral.pdf

The wavefunction is dimensionless so the action in the path integral, which of course has units of action, must be divided by a constant that also has units of action, so its dimensionless. That constant is Planks constant - its required by dimensional analysis, and of course its exact value is an experimental matter - but can, again of course, be set to 1 for theoretical convenience by a simple rescaling of units.

Thanks
Bill
 

1. What is probability amplitude?

Probability amplitude is a mathematical concept used in quantum mechanics to describe the likelihood of a quantum system transitioning from one state to another. It is represented by a complex number and is related to the probability of the system being in a particular state at a specific time.

2. What is a state operator?

A state operator, also known as a density operator, is a mathematical representation of the state of a quantum system. It is used to describe the probability distribution of a system being in a particular state, taking into account all possible states of the system.

3. How is the Galilei group related to probability amplitude and state operators?

The Galilei group is a mathematical group used to describe the symmetries of classical mechanics. It is also used in quantum mechanics to understand the relationship between probability amplitude and state operators. Specifically, the Galilei group helps to explain how these concepts are affected by time and space transformations.

4. What is the difference between classical and quantum probabilities?

Classical probabilities are based on the concept of randomness and the likelihood of an event occurring. In contrast, quantum probabilities are described by the probability amplitude of a quantum system, which takes into account both the physical state of the system and the observer's knowledge of the system.

5. How are probability amplitude, state operators, and the Galilei group used in practical applications?

These concepts are used extensively in the field of quantum mechanics, which has many practical applications such as quantum computing, cryptography, and precision measurements. They also play a crucial role in understanding and predicting the behavior of subatomic particles and systems.

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