Question about Schrödinger equation interpretation

[prehisto]

Homework Statement

Im reading and thinking about the interpretation about Schrödinger equation s solutions (wave functions) - what they really mean.
What does the amplitude of wave function correspond to?
Does it mean that if amplitude is greater then energy of particle is greater as well ?
I would appreciate any help.

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

I'm reading and thinking about the interpretation about Schrödinger equation s solutions (wave functions) - what they really mean.

Depends what you mean by "really" ;)

What does the amplitude of wave function correspond to?

It corresponds to the probability amplitude ... it is it's own thing and does not have a physical meaning by itself. You know that the wave-function, along with it's complex conjugate, form the probability density function for the variable concerned. When you start out you normally have position wave-functions.

So $\psi(x)$ is the solution to Schrodinger's equation in x.
$\psi^\star(x)\psi(x)$ is the probability density function for the particle in x.
$\int_a^b\psi^\star(x)\psi(x) dx$ is the probability of finding the particle with x value between a and b.
Only this last one has a physical meaning ... the rest are steps in a calculation.

Does it mean that if amplitude is greater then energy of particle is greater as well ?

No. The amplitude is related to probability. In general, the energy is related to the number of anti-nodes in the wave-function ... in free-space, that would be proportional to the frequency.

 

[prehisto]

Ok,thanks.
Yes, I was thinkinkg whether the amliptude has a physical interpretation.

 

[Simon Bridge]

Only natural.
It is no more physical than the magnitude of a classical probability density.
Technically it is more abstract than that - being a step removed. Some people find it a bit spooky - that such abstract ideas can have a physical impact. You'll find some people almost view the abstract math has having a deeper reality than the stuff you can measure.

Have a look at:
http://vega.org.uk/video/subseries/8
... for a glimpse of how things work.

 

[paranormal]

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of particles at the microscopic level. Its solutions, known as wave functions, represent the probability of finding a particle in a certain state at a given time. The amplitude of the wave function corresponds to the square root of the probability density of finding the particle in a particular position. Therefore, a larger amplitude does not necessarily mean a higher energy of the particle. The energy of a particle is determined by the Hamiltonian operator, which acts on the wave function and produces a value known as the expectation value of energy. This value can be different for different states of the particle, regardless of their amplitudes. In summary, the amplitude of a wave function represents the probability of finding a particle in a certain position, not its energy. I hope this helps clarify your understanding of wave function interpretation.