Just a quick question in case anybody knows… the complex amplitude value that each point in configuration space contributes to the amplitude of some future point in configuration space… the phase can be any value, but is the magnitude variable too? Or are there just lots of vector additions of the same sized vectors pointing in different directions?
As pragmatist says, the magnitude can vary, but note that the integral of the squared magnitude over the whole of phase space is constant—this is the unitarity condition, also known as “conservation of probability”. So the magnitudes cannot vary completely freely.
The “conservation of probability” only applies in low energy limits where the numbers of particles are conserved. So if there are 6 electrons, the integral over all (problem) space of the magnitude of the electron wave function will add up to 6.
But in problems where the energies are high enough, positron-electron pairs can be created from the vacuum. In problems like this, the total number of electrons is variable, is uncertain, and the “conservation of probability” does not apply.
What Alejandro said. I phrased myself carefully; “the whole of phase space” can in some cases include numbers of particles as one of the variables. Unitarity still holds.
This seems wrong to me. If you are in this high-energy case, then you should use QFT and the argument of your wavefunction is a field configuration, not particle positions. And conservation of probability still applies, in the sense that the wavefunction for the quantum field evolves unitarily.
It is just that the observable “number of particles” is not conserved. In most states, it does not even have a defined value (i.e. you can get superpositions of states with different number of particles). But properly defined, unitarity is still valid, indeed it is one of the cornerstones of quantum field theory.
The magnitude is variable as well. A wavefunction is a map from configuration space to the entire complex plane, not just the unit circle on the complex plane.
Indeed, and I might add, since the probability of an outcome is the modulus squared of the wavefunction, if the range of the wavefunction was just the unit circle, then all outcomes would always have equal probability.
Just a quick question in case anybody knows… the complex amplitude value that each point in configuration space contributes to the amplitude of some future point in configuration space… the phase can be any value, but is the magnitude variable too? Or are there just lots of vector additions of the same sized vectors pointing in different directions?
As pragmatist says, the magnitude can vary, but note that the integral of the squared magnitude over the whole of phase space is constant—this is the unitarity condition, also known as “conservation of probability”. So the magnitudes cannot vary completely freely.
The “conservation of probability” only applies in low energy limits where the numbers of particles are conserved. So if there are 6 electrons, the integral over all (problem) space of the magnitude of the electron wave function will add up to 6.
But in problems where the energies are high enough, positron-electron pairs can be created from the vacuum. In problems like this, the total number of electrons is variable, is uncertain, and the “conservation of probability” does not apply.
What Alejandro said. I phrased myself carefully; “the whole of phase space” can in some cases include numbers of particles as one of the variables. Unitarity still holds.
This seems wrong to me. If you are in this high-energy case, then you should use QFT and the argument of your wavefunction is a field configuration, not particle positions. And conservation of probability still applies, in the sense that the wavefunction for the quantum field evolves unitarily.
It is just that the observable “number of particles” is not conserved. In most states, it does not even have a defined value (i.e. you can get superpositions of states with different number of particles). But properly defined, unitarity is still valid, indeed it is one of the cornerstones of quantum field theory.
The magnitude is variable as well. A wavefunction is a map from configuration space to the entire complex plane, not just the unit circle on the complex plane.
Indeed, and I might add, since the probability of an outcome is the modulus squared of the wavefunction, if the range of the wavefunction was just the unit circle, then all outcomes would always have equal probability.