Interpretation Schrödinger's formulae

In summary, the two terms have different meanings and the classical velocity is in the direction where the j has global max at a given t. Empirical experiments verify this formulae is ok.
  • #1
Raparicio
115
0
Dear Friends,

Does anybodi knows the meaning, or anything related to the term:

[tex] \Psi \nabla \Psi^* [/tex]

or

[tex] \Psi \nabla \Psi^* - \Psi^* \nabla \Psi [/tex]

Is the representation of something in the reality?

Best reggards.
 
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  • #2
If you have a particle having a wavefunction [itex]\psi(\vec r, t)[/itex], then:

[tex]\vec J(\vec r,t)=\frac{\hbar}{2mi}(\psi^* \nabla \psi-\psi\nabla \psi^*)[/tex]

is the so-called probability current. It represents the flow of probability density, like electrical current is the flow of charge density.
 
  • #3
Spinor?

Galileo said:
If you have a particle having a wavefunction [itex]\psi(\vec r, t)[/itex], then:

[tex]\vec J(\vec r,t)=\frac{\hbar}{2mi}(\psi^* \nabla \psi-\psi\nabla \psi^*)[/tex]

is the so-called probability current. It represents the flow of probability density, like electrical current is the flow of charge density.


Thanks, Galileo, but I'm trying to "imagine" what is, for example, one of the 2 terms:

[tex] \psi^* \nabla \psi[/tex]

Has it any meaning? Is a rotor of the nabla operator?

:smile:
 
  • #4
Nope,it's just the ~ to the integral nucleus of the momentum operator.

Daniel.
 
  • #5
Galileo said:
If you have a particle having a wavefunction [itex]\psi(\vec r, t)[/itex], then:

[tex]\vec J(\vec r,t)=\frac{\hbar}{2mi}(\psi^* \nabla \psi-\psi\nabla \psi^*)[/tex]

is the so-called probability current. It represents the flow of probability density.

Sorry if this is really basic, I'm no quantum guru yet :tongue:, but could you say that the (classical) velocity is in the direction where [tex]\vec J[/tex] has global max at a given [tex]t[/tex]? Is it possible somehow to calculate [tex]\vec v[/tex] from [tex]\vec J[/tex]?
 
Last edited:
  • #6
"(Classical) velocity" has nothing to do with the probability current density...

Daniel.
 
  • #7
empirical

empirical experiments verify this formulae is ok?
 
  • #8
Which formulae...?We can't measure [itex] \vec{j}\left(\vec{r},t\right) [/itex],but only probabilities.

Daniel.
 
  • #9
Chr

dextercioby said:
Which formulae...?We can't measure [itex] \vec{j}\left(\vec{r},t\right) [/itex],but only probabilities.

Daniel.

Daniel,

I mean that if exists any experiment or example in real word that confirms that formula or some of its components. For example, if the probability to find a particle in some place or time has this formula...
 
  • #10
I'm not an experimentalist and never will be,but i can assure that this simple part of QM has been fully checked and confirmed.We can't measure certain abstract things.Since QM is a probabilistic theory,all we can do is statistics.

Daniel.
 
  • #11
dextercioby said:
I'm not an experimentalist and never will be,but i can assure that this simple part of QM has been fully checked and confirmed.We can't measure certain abstract things.Since QM is a probabilistic theory,all we can do is statistics.

Daniel.

tks Daniel.
 

What is Schrödinger's formulae?

Schrödinger's formulae, also known as the Schrödinger equation, is a mathematical equation that describes the behavior of quantum particles, such as electrons, in a given system. It was developed by Austrian physicist Erwin Schrödinger in 1926 and is a fundamental component of quantum mechanics.

What does Schrödinger's formulae tell us?

Schrödinger's formulae provides a way to calculate the probability of finding a quantum particle in a particular state or location. It describes the evolution of a quantum system over time and allows us to make predictions about the behavior of particles at the microscopic level.

What is the significance of Schrödinger's formulae?

Schrödinger's formulae is a cornerstone of quantum mechanics and has been instrumental in our understanding of the behavior of particles at the atomic and subatomic level. It has led to advancements in fields such as chemistry, materials science, and technology, and continues to be a subject of study and research in the scientific community.

How is Schrödinger's formulae used in practical applications?

Schrödinger's formulae is used in various fields, such as quantum chemistry, to predict the behavior of atoms and molecules. It is also used in the development of new technologies, such as transistors and lasers, which rely on quantum mechanics for their functioning. Additionally, it has applications in quantum computing, where it is used to simulate and solve complex problems that are not feasible with classical computers.

Are there any limitations to Schrödinger's formulae?

While Schrödinger's formulae has been incredibly successful in predicting the behavior of quantum systems, it does have its limitations. It does not take into account relativistic effects, which become significant at high speeds. Additionally, it cannot be used to describe certain phenomena, such as the collapse of the wave function, which requires the use of other equations and theories in quantum mechanics.

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