- #1
genneth said:You can't just divide by the wavefunction, so I'm afraid it's not quite correct.
pam said:It looks to me that your answer is correct if |psi> is an eigenstate of H.
It is also correct if you expand |psi> in eigenstates of H.
What other cases are there?
genneth said:You can't just divide by the wavefunction, so I'm afraid it's not quite correct.
Vagant said:|psi> is a linear combination of eigenstates of H. So, I can divide by |psi> in this case? Is such division - from (1) to (2), valid?
genneth said:Like I said, for any Hamiltonian, if you know the eigenvectors and eigenvalues, you can express an arbitrary wavefunction as a linear combination:
Given the states [tex]\left|\psi_n\right>[/tex] where [tex]\hat{H}\left|\psi_n\right>=E_n \left|\psi_n\right>[/tex], you can write:
[tex]\left|\Psi\right> = \sum_n c_n \left|\psi_n\right>[/tex].
The time evolution of the eigenstates are easy:
[tex]i\hbar \frac{d\left|\psi_n(t)\right>}{dt} = \hat{H}\left|\psi_n(t)\right> = E_n \left|\psi_n(t)\right>[/tex]
which has solutions: [tex]\left|\psi_n(t)\right> = e^{-iE_n t/\hbar} \left|\psi_n(0)\right>[/tex], because [tex]E_n[/tex] is just a number and not an operator, so you can use the usual algebraic manipulations.
Finally, since the Schroedinger equation is linear, you have that:
[tex]\left|\Psi(t)\right> = \sum_n c_n \left|\psi_n(t)\right> = \sum_n c_n e^{-iE_n t/\hbar} \left|\psi_n(0)\right>[/tex].
Note that crucially you need to be able to find the eigenstates and eigenvalues of the Hamiltonian, which is usually quite hard.
The time-dependent Schrödinger equation is a fundamental equation in quantum mechanics that describes the evolution of a quantum system over time. It is used to calculate the wave function of a quantum system, which contains all the information about the system's state.
The time-dependent Schrödinger equation is typically solved using numerical methods or analytical techniques such as separation of variables. The solution yields the wave function of the quantum system at any given time.
The time-dependent Schrödinger equation is used in various fields, including quantum chemistry, atomic and molecular physics, and solid state physics. It is essential for understanding the behavior of quantum systems and predicting their properties.
The time-dependent Schrödinger equation is limited to non-relativistic quantum systems. It also assumes that the system is in a pure state and does not take into account any external influences or interactions with other systems.
The square of the wave function, which is the solution to the time-dependent Schrödinger equation, represents the probability of finding a particle in a particular location at a given time. This interpretation is known as the Born rule and is a fundamental principle of quantum mechanics.