- #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
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