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theoriginal169
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You can think about the wave function as a classical field. It assigns a complex number to every point on your space under investigation, let say the real line.

 

[math]\psi : \mathbb{R} \rightarrow \mathbb{C}[/math].

 

(More geometrically you can think of it as a section of a line bundle, this bundle may not be trivial in a more general setting. But this won't worry us now.)

 

From this complex number you build a probability of the particle to be found at that point. We define a probability density as

 

[math]|\psi(x)|^{2}[/math] (used the complex modulus here).

 

Then probability of a measurement of the particle's position yielding a value in the interval [math][a,b][/math] is

 

[math]P(a,b) = \int^{b}_{a}dx |\psi(x)|^{2}[/math],

 

were we have assume the natural normalisation

 

[math]1 = \int^{+\infty}_{- \infty}dx |\psi(x)|^{2}[/math].

 

Hope that is of some help.

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is it a probibilty densty related to time?

 

that depends entirely on the wave function you are talking about. Waves on water (obviously) have a wave function for example. In this case it is a surface amplitude related to distance and time. Light has a wave function which describes the electric and magnetic fields as a function of position and time. In essence it is a deviation from the average value (height, field strength or whatever) with distance and time. In the case of QM it is a deviation on the complex plane.

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is it a probibilty densty related to time?

 

In general yes it is a function of time (in the Schrodinger representation). However we can formally* separate it as

 

[math]\Psi(t,x) = U(t) \psi(x)[/math]

 

where the time-evolution operator is given by

 

[math]U(t) = \exp \left( - \frac{i H t}{\hbar}\right)[/math],

 

here [math]H[/math] is the Hamiltonian operator.

 

The wave function [math]\Psi[/math] satisfies the time-dependant Schrodinger equation and [math]\psi[/math] satisfies the time-independent Schrodinger equation. Have a look at the Wiki article.

 

 

 

* I will neglect any issues of convergence for unbounded operators in the exponential. So this separation will in general be understood very formally.

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q(x,t) how does it works dynamical variable in wave function expectation value of p


Merged post follows:

Consecutive posts merged
In general yes it is a function of time (in the Schrodinger representation). However we can formally* separate it as

 

[math]\Psi(t,x) = U(t) \psi(x)[/math]

 

where the time-evolution operator is given by

 

[math]U(t) = \exp \left( - \frac{i H t}{\hbar}\right)[/math],

 

here [math]H[/math] is the Hamiltonian operator.

 

The wave function [math]\Psi[/math] satisfies the time-dependant Schrodinger equation and [math]\psi[/math] satisfies the time-independent Schrodinger equation. Have a look at the Wiki article.

 

 

 

* I will neglect any issues of convergence for unbounded operators in the exponential. So this separation will in general be understood very formally.

Hamiltonian operator what does it do derivatie and x -h2i

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Hamiltonian operator what does it do derivatie and x -h2i

 

In classical mechanics the Hamiltonian is the "energy function", it is the sum of the Kinetic and Potential energy. In quantum mechanics is it promoted to an operator on the Hilbert space of states.

 

It's importance in both classical and quantum mechanics is that it describes the time evolution of the system.

 

In classical mechanics we have Hamilton's equations which describe how canonical coordinates evolve (we have a flow on the phase space).

 

In quantum mechanics we have Heisenberg equations which describe the evolution of the observables (or more general operators if we wish). Here we think of the states as being time independent and the operators as time dependant.

 

As an aside the Stone--von Neumann theorem which states that for finite degrees of freedom the Schrodinger and Heisenberg representations of the canonical commutation relations are unitary equivalent. In essence, we have the operator [math]U[/math] which I defined earlier. So we can without any loss of generality consider either representation. There is also the "mixed" Dirac or interaction picture, which is useful for time-dependant Hamiltonians and scattering theory.

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