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alimehrani
- 6
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why it is not possible to normalize the free-particle wawe functions over the whole range of motion of the particles?
tom.stoer said:Let me see if I understood you correctly.
You propose to use (nearly) arbitrary wave packets constructed from plane waves (Fourier modes) and let these wave packets evolve in time using the free Hamiltonian. That means you construct normalizable wave functions, but they are no longer eigenstates of the free Hamiltonian.
Yes, of course you are right. I assumed that "free particle" means "eigenstate of the free Hamiltonian H", but that need not be the case.
@alimehrani: what was your intention?
A free particle is a theoretical concept in physics that refers to a particle that is not influenced by any external forces or interactions. In other words, it moves freely in space without any constraints or obstacles.
Normalization is a mathematical procedure used to ensure that the total probability of finding a free particle at any point in space is equal to 1. This allows for the calculation of the probability of finding the particle within a specific region of space.
Normalization is important in quantum mechanics because it ensures that the wave function, which describes the behavior of a free particle, is physically meaningful. It also allows for the calculation of measurable quantities, such as the probability of finding the particle in a specific location.
The normalization of a free particle is calculated by integrating the square of the wave function over all space and equating it to 1. This involves using mathematical techniques, such as the Fourier transform, to solve the Schrödinger equation and obtain the wave function.
The units of free particle normalization depend on the specific system being studied. In general, it has the units of inverse length squared, such as meters-2 or angstroms-2. This is because the wave function, which is squared in the normalization integral, has units of inverse length.