Describing Energy for Time-Dependent States

[Isaac0427]

Hi all! Sorry if this question is stupid. The the time-independent Schrödinger equation describes energy for a time-independent system, and the time-dependent Schrödinger equation describes the time evolution of the wavefunction. So, how would you describe the energy for a time-dependent system? Thanks!

 

[blue_leaf77]

It's in general very difficult to describe the exact time evolution of the wavefunction in the case of time-dependent Hamiltonian. However, there is one special case, which is the two-level system interacting with a sinusoidal electromagnetic field, which can be solved exactly. In this system, the system's state alternates between the two levels in a period determined by the EM field's oscillation. This oscillation of states is also sometimes called Rabi flopping.

 

[Isaac0427]

It's in general very difficult to describe the exact time evolution of the wavefunction in the case of time-dependent Hamiltonian. However, there is one special case, which is the two-level system interacting with a sinusoidal electromagnetic field, which can be solved exactly. In this system, the system's state alternates between the two levels in a period determined by the EM field's oscillation. This oscillation of states is also sometimes called Rabi flopping.

Ok, but how would you describe the energy of a time-dependent wavefunction (please excuse my bad terminology).

 

[Isaac0427]

Hold on, would iħ∂/∂t characterize time evolution and energy?

 

[atyy]

Within the orthodox interpretation, quantum systems do not have properties until they are measured. So a quantum system does not have an energy until its energy is measured. When the energy is measured, the outcome is random and distributed according to the Born rule.

 

[Isaac0427]

Within the orthodox interpretation, quantum systems do not have properties until they are measured. So a quantum system does not have an energy until its energy is measured. When the energy is measured, the outcome is random and distributed according to the Born rule.

Ok, so then what does the time-independent Schrödinger equation describe?

EDIT:
Also, particles don't have defined properties, which I know, but I also know that the wavefunction can describe the expectation value of a specific property. Using the hamiltonian operator, you can find the expectation value of energy, correct?

 

[Nugatory]

Using the hamiltonian operator, you can find the expectation value of energy, correct?

yes.

 

[Nugatory]

Ok, so then what does the time-independent Schrödinger equation describe?

Its eigenvalues are the values that you might find for the energy when and if you measure it.

 

[Isaac0427]

yes.

Could you say for a time-dependent system that $\hat H \psi = \hat E \psi = E \psi$ where $\hat H = \frac{- \hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} + V$ and $\hat E = i \hbar \frac{\partial}{\partial t}$?

 

[kith]

The the time-independent Schrödinger equation describes energy for a time-independent system [...]

This needs some elaboration.

In fact, I think that the very name "time-independent Schrödinger equation" is bad terminology. A better name is "eigenvalue equation for the Hamilton operator". As Nugatory wrote, its solutions are the possible energies and associated wavefunctions which you get if you perform an energy measurement on your system or if you prepare your system in a state of definite energy. So the Hamilton operator is the operator which is associated with energy in QM.

If your system is in such an energy eigenstate, the (time-dependent) Schrödinger equation tells you that it doesn't change with time.

 

[kith]

Could you say for a time-dependent system that $\hat H \psi = \hat E \psi = E \psi$

The last identity is false in general, i.e. it only holds if \psi is a state of definite energy.

 

[Isaac0427]

The last identity is false in general, i.e. it only holds if ψψ\psi is a state of definite energy.

Why? If $\hat H \psi = E \psi$ (time independent Schrödinger equation) and $\hat H \psi = \hat E \psi$ (time dependent Schrödinger equation), then why is $\hat E \psi = E \psi$ not true for time-dependent systems? Is it because $\hat H \psi = E \psi$ does not apply to time-dependent systems? Does it?

 

[kith]

Why? If $\hat H \psi = E \psi$ (time independent Schrödinger equation) and $\hat H \psi = \hat E \psi$ (time dependent Schrödinger equation), then why is $\hat E \psi = E \psi$ not true for time-dependent systems? Is it because $\hat H \psi = E \psi$ does not apply to time-dependent systems?

Using your terminology, the answer is yes. But dividing systems into time-dependent and time-independent is problematic. It is the wavefunction (or speaking in more general terms, the state) of a system which does or doesn't change over time.

As I tried to tell you in my post #10, the (time-dependent) Schrödinger equation and the eigenvalue equation for the Hamiltonian operator are conceptually very different.

The (time-dependent) Schrödinger equation tells you how the wavefunction of your system changes over time. This equation is always valid (even in relativistic quantum theory, if we use the appropriate generalization). If you have an arbitrary system with an arbitrary initial wavefunction, the (time-dependent) Schrödinger equation tells you -in principle- how it will evolve in time.

The eigenvalue equation for the Hamiltonian on the other hand doesn't correspond to a certain physical situation. It tells you which definite energies are possible for your system and how the corresponding wavefunctions look like. In QM, such an equation exists for for every observable quantity (like momentum, angular momentum, spin, etc.) and the Hamiltonian corresponds to the observable "energy".

 

[blue_leaf77]

Ok, but how would you describe the energy of a time-dependent wavefunction (please excuse my bad terminology).

By calculating the expectation value of the Hamiltonian. I think you have noticed this fact and also Nugatory has confirmed this for you.

Why? If $\hat H \psi = E \psi$ (time independent Schrödinger equation) and $\hat H \psi = \hat E \psi$ (time dependent Schrödinger equation), then why is $\hat E \psi = E \psi$ not true for time-dependent systems? Is it because $\hat H \psi = E \psi$ does not apply to time-dependent systems? Does it?

I don't think you can find $\psi$ such that $\hat{H}(t) \psi = E\psi$. In the case of time-independent Hamiltonian, the time evolution for an energy eigenstate $u_n(t)$ is described by $e^{-iE_n t/\hbar}$. Thus, $\hat{E} u_n(t)$ reduces to $E_n u_n(t)$. But for time-dependent system, the term describing the time evolution is no longer as simple-looking as $e^{-iE_n t/\hbar}$.

 

[PeroK]

Why? If $\hat H \psi = E \psi$ (time independent Schrödinger equation) and $\hat H \psi = \hat E \psi$ (time dependent Schrödinger equation), then why is $\hat E \psi = E \psi$ not true for time-dependent systems? Is it because $\hat H \psi = E \psi$ does not apply to time-dependent systems? Does it?

Leaving aside the physics, you are not using the mathematics carefully enough.

The time-independent equation emerges from the technique of separation of variables, where you look for $\Psi(x, t) = \psi(x) f(t)$. If the potential, hence the Hamiltonian, is time-independent, then the equation separates and the spatial equation is the "time independent" equation. In this case, $\psi$ is a function of the spatial coordinates and you get an eigenvalue equation involving this function as the eigenfunction:

$\hat {H} \psi(x) = E \psi(x)$

If the potential is time-dependent, then the Schroedinger equation does not separate and you don't get a time-independent equation. Moreover, the concept of eigenvalues doesn't make any mathematical sense. You have an operator equation that does not correspond to eigenvalues of any function:

$ih \hat{D}_t \Psi(x, t) = \hat {H}(t) \Psi(x, t)$

This is a more general operator equation on functions of two variables $(x, t)$ and there is no way to introduce an eigenvalue here at all.

 

[Isaac0427]

By calculating the expectation value of the Hamiltonian. I think you have noticed this fact and also Nugatory has confirmed this for you.

Ok, but now I am getting very different answers. Does the equation $\hat H \psi = E \psi$ hold for time dependent systems? If not, how would you describe the expectation eigenalue of energy $E$?

 

[blue_leaf77]

Ok, but now I am getting very different answers. Does the equation $\hat H \psi = E \psi$ hold for time dependent systems? If not, how would you describe the expectation eigenalue of energy $E$?

Take a look at the last paragraph of post #14 as well as the second half of post #15.

 

[Isaac0427]

Take a look at the last paragraph of post #14 as well as the second half of post #15.

Ok, so you are saying that you can't do it with a time dependent system. However, in an earlier post you said

By calculating the expectation value of the Hamiltonian. I think you have noticed this fact and also Nugatory has confirmed this for you.

As an answer to a question about describing the energy of a time dependent wavefunction. Also, if you can't have an energy eigenvalue for time dependent systems, yet energy is an observable quantity, wouldn't the operator postulate (with every observable quantity q there is an operator Q...) be violated?

 

[blue_leaf77]

However, in an earlier post you said
As an answer to a question about describing the energy of a time dependent wavefunction.

What's wrong with calculating an expectation value of energy? You can do it regardless of whether the Hamiltonian depends on time.

Also, if you can't have an energy eigenvalue for time dependent systems, yet energy is an observable quantity, wouldn't the operator postulate (with every observable quantity q there is an operator Q...) be violated?

The eigenvalue problem can still be applicable in the case of time-dependent Hamiltonian, but it will be time-specific, meaning that at each instant of time $t'$, the Hamiltonian $H(t')$ will have a specific spatial arrangement, and you can therefore write an eigenvalue problem exclusively for this time $H(t') u_n^{t'} = E_n^{t'} u_n^{t'}$. As soon as, an infinitesimal change in time occurs, the Hamiltonian will change and therefore another different eigenvalue problem applies.

 

[vanhees71]

Ok, but now I am getting very different answers. Does the equation $\hat H \psi = E \psi$ hold for time dependent systems? If not, how would you describe the expectation eigenalue of energy $E$?

No, energy eigenstates represent stationary states. Note that it is not the Hilbert space vector $|\psi \rangle$ that represents the state but the ray it represents or, equivalently, the projector $\hat{\rho}_{\psi} = |\psi \rangle \langle \psi |$ as the statistical operator.

For a time-dependent state, the energy is indetermined, because it's not an energy eigenstate.

 

[PeroK]

Ok, but now I am getting very different answers. Does the equation $\hat H \psi = E \psi$ hold for time dependent systems? If not, how would you describe the expectation eigenalue of energy $E$?

You need an example of a solution for a time-dependent potential. If you have Griffith's book, he gives one for a moving delta-function well. Problem 2.50.

If you apply the Hamiltonian (or, equivalently, apply the time differential operator), you get:

$\hat{H}\Psi = i\hbar \hat{D}_t \Psi = (E + \beta(x, t)) \Psi$

Where $E$ is the expected value of the Hamiltonian and $\beta(x, t)$ is an additional function that integrates to 0 with $|\Psi|^2$.

This example shows that, as explained above, the concept of the expected energy being an eigenvalue doesn't generalise to time-dependent systems.

 

[Isaac0427]

The eigenvalue problem can still be applicable in the case of time-dependent Hamiltonian, but it will be time-specific, meaning that at each instant of time $t'$, the Hamiltonian $H(t')$ will have a specific spatial arrangement, and you can therefore write an eigenvalue problem exclusively for this time $H(t') u_n^{t'} = E_n^{t'} u_n^{t'}$. As soon as, an infinitesimal change in time occurs, the Hamiltonian will change and therefore another different eigenvalue problem applies.

Thank you very much, this makes so much more sense.

 

[PeroK]

The eigenvalue problem can still be applicable in the case of time-dependent Hamiltonian, but it will be time-specific, meaning that at each instant of time $t'$, the Hamiltonian $H(t')$ will have a specific spatial arrangement, and you can therefore write an eigenvalue problem exclusively for this time $H(t') u_n^{t'} = E_n^{t'} u_n^{t'}$. As soon as, an infinitesimal change in time occurs, the Hamiltonian will change and therefore another different eigenvalue problem applies.

I must confess, I don't see how those eigenvalues relate to the time-dependent system. In the case of the moving delta well, at each time $t'$ you would get the same single bound state energy $E_0$.

But this energy isn't the energy of the bound state of the time-dependent system, which is $E_0 + \frac{1}{2}mv^2$.

There is not necessarily a solution of the time-dependent Schroedinger equation that reduces to $u_n^{t'}$ at time $t'$. In the case of the moving well, the time-dependent solution has an additional term in $x$ that is not present in any of the fixed-time solutions. In other words, you can't solve a PDE in two variables by solving the PDE in $x$ for each fixed time $t'$.

 

[blue_leaf77]

In the case of the moving delta well, at each time t′t′t' you would get the same single bound state energy E0E0E_0.

The energy may be the same but the eigenstate is clearly different for different $t'$.

In the case of the moving well, the time-dependent solution has an additional term in xxx that is not present in any of the fixed-time solutions.

Can you please mention the particular problem in Griffith's book which discusses this problem?

In other words, you can't solve a PDE in two variables by solving the PDE in xxx for each fixed time t′t′t'.

Yes, indeed. The resulting equation will be of only one variable because the time has been fixed.

 

[blue_leaf77]

I must confess, I don't see how those eigenvalues relate to the time-dependent system. In the case of the moving delta well, at each time $t'$ you would get the same single bound state energy $E_0$.

But this energy isn't the energy of the bound state of the time-dependent system, which is $E_0 + \frac{1}{2}mv^2$.

There is not necessarily a solution of the time-dependent Schroedinger equation that reduces to $u_n^{t'}$ at time $t'$. In the case of the moving well, the time-dependent solution has an additional term in $x$ that is not present in any of the fixed-time solutions. In other words, you can't solve a PDE in two variables by solving the PDE in $x$ for each fixed time $t'$.

I think I should have given more thought on this. When I wrote post #20, I was having an example of a step potential in time, e.g. $V(x) = V_1$ for $t<0$ and $V(x) = V_2$ for $t>0$, in mind. In such problem, one can separate the Schroedinger equation into two parts: before and after $t=0$. In each time period, one can solve the trivial time-independent equation.
The reason I force that argument with a fixed time $t'$ is that because I was trying to connect the idea about the possible energy measurement at each instant of time as this is an essential notion in quantum mechanics. Namely, each measurement at any instant of time will in general have some probable values. I was thinking that, if it's not possible to construct an eigenvalue problem at any instant of time, then how one would be able to determine the possible outcomes of the energy measurement during that time.

 

[PeroK]

The problem in Griffiths is:

$V(x, t) = -\alpha \delta(x - vt)$

To which he provides the exact solution:

$\Psi(x, t) = \frac{\sqrt{m\alpha}}{\hbar} exp(-m\alpha |x - vt|/\hbar^2) exp(-i[(E_0 + \frac{1}{2}mv^2)t -mvx]/\hbar)$

Where $E_0 = \frac{-m\alpha^2}{2\hbar^2}$ is the bound state energy of the stationary delta function well.

The expected energy of this solution is $E_0 + \frac{1}{2}mv^2$

If you take the potential at any time $t'$, it's just a fixed delta potential well at position $vt'$ with a single bound energy $E_0$. Hence, you don't get the solution or energy values for the time-dependent equation from the series of fixed-time solutions.