What is wavefunction in the time-dependent schrodinger equation?

[goodphy]

Hello.

The wave function or state vector (callled 'Ket') ψ in the time-dependent schrodinger equation

$i\hbar\frac{∂ψ}{∂t}=\widehat{H}ψ$

is the just energy eigenfunction or any wavefunction for the given system?

For example, can ψ be momentum eigenfunction or angular momentum eigenfunction, etc?

 

[kith]

The time-dependent Schrödinger equation holds for arbitrary state vectors / functions. The time-independent Schrödinger equation holds only for eigenstates of the Hamiltonian.

 

[dextercioby]

The time-dependent Schrödinger equation holds for arbitrary state vectors / functions. [...]

Since it's an equation, not a mere equality, it holds for its set of solutions only. That 'arbitrary' is wrongly placed there.

 

[goodphy]

Since it's an equation, not a mere equality, it holds for its set of solutions only. That 'arbitrary' is wrongly placed there.

Alright thus...could you give me the idea about what kinds of solutions are holding for the time-dependent Schroedinger equation?

 

[HAMJOOP]

You can try the method "seperation of variable"
For simplicity, we stick in 1-D

Step 1 let ψ(x,t) = X(x)T(t)
now the equation(PDE) becomes ODE (second order)

Step 2 divide both sides by X(x)T(t)
then you should get LHS(depends on t only) = RHS(depends on x only)
therefore LHS = RHS = constant = λ

Step 3 for different λ, you will have different solution.


ψ is an eigenfunction of Hamiltonian ⇔ uncertainty of energy is zero (energy is also quantized)
That is ψ is a stationary state

It is possible to have angular momentum eigenfunction (in central force field)

However it is not possible to have momentum eigenfunction.
Coz it leads to zero uncertainty in momentum which contradicts the uncertainty principle

 

[goodphy]

You can try the method "seperation of variable"
For simplicity, we stick in 1-D

Step 1 let ψ(x,t) = X(x)T(t)
now the equation(PDE) becomes ODE (second order)

Step 2 divide both sides by X(x)T(t)
then you should get LHS(depends on t only) = RHS(depends on x only)
therefore LHS = RHS = constant = λ

Step 3 for different λ, you will have different solution.


ψ is an eigenfunction of Hamiltonian ⇔ uncertainty of energy is zero (energy is also quantized)
That is ψ is a stationary state

It is possible to have angular momentum eigenfunction (in central force field)

However it is not possible to have momentum eigenfunction.
Coz it leads to zero uncertainty in momentum which contradicts the uncertainty principle

Thus are you saying that the time-dependent Schroedinger equation has solution which is eigen function of the Hamiltonian and angular momentum eigenfunction is also possible solution for this equation since the angular momentum operator is commuted with Hamiltonian?

 

[Avodyne]

Assuming the hamiltonian is time independent, the general solution of the time-dependent Schroedinger equation is
$$\psi(x,t)=\sum_n c_n\psi_n(x)e^{-iE_n t/\hbar}$$where $\psi_n(x)$ is an energy eigenstate,
$$\hat H\psi_n(x)=E_n\psi_n(x)$$and $c_n$ is an arbitrary complex number.