Question about schrodinger equation

[stunner5000pt]

Note that this is not homework... I am just curious

The time independent Schrödinger equation can be written as
$$\hat{H} \psi = E\psi$$


IS there ever a time taht the above equation is not true??

what about the time independent case?? We haven't gone over that in class so I am not quite sure about the cases.

Thanks for your input!

 

[vanesch]

Note that this is not homework... I am just curious

The time independent Schrödinger equation can be written as
$$\hat{H} \psi = E\psi$$


IS there ever a time taht the above equation is not true??

The time-independent SE gives you those specific states that are stationnary. If you're looking for those states, then that equation is for you
It is not a general solution to a quantum problem, but looks as specific kinds of cases. However, it also turns out that the solutions of this equation (which is simply the eigen-equation of the hamiltonian in fact) allows you to find the more general time-dependent solutions.

As an analogy, consider it something like the static equilibrium equation F = 0 in Newtonian physics. It is not the general dynamical solution dp/dt = F, but only gives you the solutions of static equilibrium. However, here, it is less clear how the static solutions can help you find the general dynamic solutions, while in the quantum case, there is a clear link.

 

[masudr]

As vanesch says, it only satisfies the stationary states. What it does is gives you the states of definite energy (which are useful for several reasons), in this case the energy is E.

Compare with

$$\hat{p}\psi=p\psi$$

which works for states of definite momentum, specifically of momentum p.

Both of these are the eigenvalue equations for the respective operators. The time dependent equation (the Schrödinger equation proper) is given by

$$\hat{H}| \psi \rangle = i\hbar \frac{d}{dt} | \psi \rangle$$

and this holds for any quantum state. One can almost (problems arise with spin-statistics) view this equation as defining which $| \psi \rangle$ are allowable quantum states. In the case of time-independent Hamiltonians, we can separate the time dependence out, as I'm sure you will soon find out in class.

 

[Jheriko]

The answers above are great, but here are some shorter ones.

$$\hat{H} \psi = E\psi$$
IS there ever a time taht the above equation is not true??

Yes.

what about the time dependent case?? We haven't gone over that in class so I am not quite sure about the cases.

The above equation does not hold for time-dependent cases. i.e. when the wave function changes shape over time. A simple example is a particle moving through empty space (as a plane wave).

To make the time dependent form we replace the constant E with an expression which can handle the variations over time $\mathrm{i} \hbar \frac{\partial \psi}{\partial t}$:

$$\hat{H} \psi = \mathrm{i} \hbar \frac{\partial \psi}{\partial t}$$

AFAIK, this is identical to the form given in the previous posts with kets.

Hope this helps.

 

[stunner5000pt]

so stationary states are defined at hte square modulus of the wave function not having a time dependance.
How would we know this before solving for the wavefunction? If it was time independent then certain $\psi$ would not depend on time

would this mean taht all solutions of the TISE (time indepdnant Schrödinger equation) are stationary states?? Since the solution for the time dependence is always $\exp(-iEt/\hbar)$. If we could do that then certainly our potential is not time varying. Hence there's never a time depdance and the state is stationary always?

 

[Jheriko]

so stationary states are defined at hte square modulus of the wave function not having a time dependance.
How would we know this before solving for the wavefunction?

We don't, but if we do know then we can use that to simplify the problem. If we try and solve a dynamic problem with the time-independent equation then we should eventually end up with some nonsensical expression.

It should be quite obvious when to use which one. Since ground states shouldn't change spontaneously we can use the time-independent equation to approximate them. If we wish to model an evolving state then we need to use the time-dependent form.


I think I may have made a mistake in the formula somewhere though. I keep thinking that $\frac{\partial \psi}{\partial t}$ is equal to zero when nothing is changing over time, so that we should have $\hat{H} \psi = 0$ for the time-independent form. So, I'm not sure exactly why the E's are not zero... probably some oversight on my part...

 

[Aero]

I think I may have made a mistake in the formula somewhere though. I keep thinking that $\frac{\partial \psi}{\partial t}$ is equal to zero when nothing is changing over time, so that we should have $\hat{H} \psi = 0$ for the time-independent form. So, I'm not sure exactly why the E's are not zero... probably some oversight on my part...

Time-independence of a physical system, only requires the square modulus of the wavefunction to be time-independent, not the wave function itself. This means that the wave function itself can vary with time, although only in a way that does not change its sqaure modulus, which means that it can only change by phasing, in which case $\hat{H} \psi = E \psi$

 

[Jheriko]

Ah, of course, not sure why I forgot that, it is even mentioned already in this thread. Its also pretty damned important for calculations... oh well.

 

[stunner5000pt]

Time-independence of a physical system, only requires the square modulus of the wavefunction to be time-independent, not the wave function itself. This means that the wave function itself can vary with time, although only in a way that does not change its sqaure modulus, which means that it can only change by phasing, in which case $\hat{H} \psi = E \psi$

change the phasing...

where is the pahse factor appearing in the wave equation?
Lets say for the wavefunction of hte infinite square well

$$\Psi(x,t) = N \sin\frac{n\pi x}{a} e^{-\frac{\i E_{n}t}{\hbar}}$$

well a wave can also be written like
$$Asin(kx-\phi) + Bcos(kx-\phi)$$

i don't quite see how the phi factor comes about in the wave equation... where would it be??
could it be like this?
$$\Psi(x,t) = N \sin\left(\frac{n\pi x}{a}+\Phi\right) e^{-\frac{\i E_{n}t}{\hbar}}$$

 

[jtbell]

You can add an arbitrary constant complex phase factor to a QM wave function, without changing the physics:

$$\Psi(x,t) = N \sin\left(\frac{n\pi x}{a} \right) e^{-i \left( \frac{ E_{n}t}{\hbar} + \phi \right)}$$Physically, this corresponds to the fact that the "zero point" of potential energy (and therefore also of total energy) is arbitrary. Note what happens when you set $V(x) = V_0$ (a constant) inside the infinite square well, and then solve the Schrödinger equation again.

 

[stunner5000pt]

You can add an arbitrary constant complex phase factor to a QM wave function, without changing the physics:

$$\Psi(x,t) = N \sin\left(\frac{n\pi x}{a} \right) e^{-i \left( \frac{ E_{n}t}{\hbar} + \phi \right)}$$


Physically, this corresponds to the fact that the "zero point" of potential energy (and therefore also of total energy) is arbitrary. Note what happens when you set $V(x) = V_0$ (a constant) inside the infinite square well, and then solve the Schrödinger equation again.

then wouldn't we have to consider different situations for the solution suppose E> V0, E<V0, E=V0??

three possible solutions then right? Just want to make sure before i go ahead with this. The wavefunction would still ahve to vanish at the boundary points, though.

 

[vanesch]

You can add an arbitrary constant complex phase factor to a QM wave function, without changing the physics:

$$\Psi(x,t) = N \sin\left(\frac{n\pi x}{a} \right) e^{-i \left( \frac{ E_{n}t}{\hbar} + \phi \right)}$$Physically, this corresponds to the fact that the "zero point" of potential energy (and therefore also of total energy) is arbitrary. Note what happens when you set $V(x) = V_0$ (a constant) inside the infinite square well, and then solve the Schrödinger equation again.

Except that it won't give you a constant phase factor, but one of the form $$e^{-i V_0 t/\hbar}$$

 

[jtbell]

Oops, you're right.

I should have realized that "relocating" the bottom of the well would simply change $E_n$ to $E_n - V_0$.

The arbitrary $\phi$ simply indicates the "initial value" of the complex oscillation, and doesn't correspond to anything physical. Any time you calculate an expectation value or probability, it always contributes a net factor of $e^{i \phi} e^{-i \phi} = 1$.

This is true only for a $\phi$ that doesn't depend on x, of course. If $\phi$ depends on x, that is, the initial phase is different at different locations instead of being uniform everywhere, things get interesting!

 

[jtbell]

then wouldn't we have to consider different situations for the solution suppose E> V0, E<V0, E=V0??

In principle yes, but you already have to do that when $V_0 = 0$, i.e. consider the cases E > 0, E < 0, E = 0.

The wavefunction would still ahve to vanish at the boundary points, though.

For $E < V_0$ you'll find that it's impossible to construct a wave function that vanishes at both sides of the well. Have you done the finite square well ($V = V_0$ outside the well, and 0 inside) yet, or tunneling through a barrier (V = 0 outside and $V = V_0$ inside)? These are the situations where people usually first tackle $E < V_0$.

 

[stunner5000pt]

In principle yes, but you already have to do that when $V_0 = 0$, i.e. consider the cases E > 0, E < 0, E = 0.



For $E < V_0$ you'll find that it's impossible to construct a wave function that vanishes at both sides of the well. Have you done the finite square well ($V = V_0$ outside the well, and 0 inside) yet, or tunneling through a barrier (V = 0 outside and $V = V_0$ inside)? These are the situations where people usually first tackle $E < V_0$.

yes i have done the tunnelling cases
so if its impossible to make the wavefunction vanish at the end points then what boundary conditions could we apply??

 

[jtbell]

We don't have any choice in the boundary conditions. The boundary conditions are fixed by the choice of potential. For an infinite square well, we must require that $\psi = 0$ at the boundaries. If a general mathematical solution of the S.E. with a particular E cannot satisfy that condition, then it is not admissible as a physical solution for this situation.

As you've seen in the tunnelling case, the general solution for E < V involves real exponentials: $\psi = Ae^{kx} + Be^{-kx}$. There's no way to set the arbitrary constants A and B so that $\psi = 0$ at both boundaries of the infinite square well. Therefore it is impossible to have E < V for that situation.

 

[stunner5000pt]

Oh i have another question about this
Is this always true??
$$\hat{H} \Psi = E \Psi$$
Here we talk about the wavefunction in general
I think this is not always true ... but i don't know understand why?? I m thinking that it ahs something to do with the time dependent case where the right is
$$i\hbar \frac{d}{dt} \Psi$$
this term need not repsent the energy... right?

 

[dextercioby]

Oh i have another question about this
Is this always true??
$$\hat{H} \Psi = E \Psi$$
Here we talk about the wavefunction in general
I think this is not always true ... but i don't know understand why?? I m thinking that it ahs something to do with the time dependent case where the right is
$$i\hbar \frac{d}{dt} \Psi$$
this term need not repsent the energy... right?

You've already been explained the difference between a time-dependent Hamiltonian in Schroedinger picture and a time-independent one. And how that determins the time-evolution of the state vector.

$$\hat{H} \Psi = E \Psi$$

is a spectral equation for an operator on separable Hilbert space and nothing more. It has solutions in the Hilbert space iff the spectrum of the Hamiltonian is discreet.

Daniel.