V(x) = Vo —> Partícula lliure —> —> No normalitzable —> No pot representar un estat físic —> Necessitem una nova interpretació
Interpretació física: Paquets d’ona
—> Velocitat de grup i de fase
Interpretació matemàtica: Funció d’ona restringida en un pou de potencial infinit
—> ,
—> Cada valor propi està degenerat 2 vegades
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In contrast with the infinite well case, for the free particle system we don't have boundary conditions on the wavefunction, and therefore the energy E
can take any positive value: the only contribution to the total energy
is the kinetic energy, which is positive-definite. So in the case of the free particle system the energy is not quantised and can take a continuum of values.
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The way to find physical wavefunctions is to combine solutions with different energies. In such case, Δp will take a finite value and hence Δx can be finite as well. And if a particle is localised, one will be able to construct normalisable wave functions.
“In the case of a quantum system like the free particle, one should replace a sum by an integral.”
Most general wavefunction for a free particle —> wave packet
“Note that the wave packet itself does not have a unique energy of wavelenght.”
Using the property of ortogonalithy:
Bla bla bla, we therefore find that the function ϕ(k) is given by:
Osigui, la transformada de Fourier
—> Aahhh aleshores allo de notació correcta és només per una free particle??
Wave packet
Consider free particle with the following wave function at t=0:
Where A and a are real and positive constants. We would like to find the function $\phi(k)$ that defines the wave packet leading to $\Psi(x, 0)$ and determine how the properties of the resulting wave packet vary with time.
First of all, we want to normalize $\Psi(x, 0)$, that is, find the value of the constant $A$
Note that here I will take $A$ to be real, but we could have always multiplied it by an arbitrary complex phase.
Next we want to evaluate the function $\phi(k)$ that ensures the desired Gaussian profile of the wave packet:
Where we have used the following result for definite Gaussian integrals
Now that we have found ϕ(k) and the normalisation constant A, we can determine the time evolution of the wave packet:
