Having trouble with time (in)dependant solutions

  • Thread starter ChaosCon343
  • Start date
  • Tags
    Time
In summary, the conversation discusses the time-dependent and time-independent solutions of the Schrödinger equation in quantum mechanics. The time-independent solution is obtained by separating the space and time variables, but it is still able to reconstruct any arbitrary \Psi(x,t). This is because an electron in a potential like an infinite square well can have different \Psi's. The time-dependent solution, on the other hand, is a superposition of particular solutions and is necessary to have a time-dependence in observable quantities. The probability density is given by a combination of the individual probabilities and an interference term, which contains all the time-dependence.
  • #1
ChaosCon343
7
0
Hey all -

I'm taking my first quantum class this year, and I'm still on really shaky ground about time-dependent and time-independent solutions of the Schrödinger equation. I understand that the time independent Schrödinger equation comes from separating your space and time variables, but I have trouble interpreting the meanings of both types of solutions. Particularly, why can you reconstruct any arbitrary [tex]\Psi(x,t)[/tex] out of the time-independent solutions? Rather, why can an electron in, say, an infinite square well potential have different [tex]\Psi[/tex]'s?
 
Physics news on Phys.org
  • #2
Solve the time-dependent Schroedinger equation [tex]i\hbar\frac{\partial}{\partial t}\Psi(x,t) = H\Psi(x,t)[/tex] by separation of variables to give the following particular solutions (which have the counterintuitive property of predicting time-independent observables):

[tex]
\Psi(x,t) = \phi_E(x)e^{-\frac{i}{\hbar}Et}\;\;\;\;\;\;\;\;|\Psi(x,t)|^2 = |\phi_E(x)|^2
[/tex]

Where has the time gone? It is restored to us by a general solution to the TDSE - an arbitrary superposition of the particular solutions:

[tex]
\Psi(x,t) = \sum_{n=1}^{\infty} a_n\phi_n(x) e^{-\frac{i}{\hbar}E_nt} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{\mathrm{(discrete\;spectrum)}}
[/tex]

[tex]
\Psi(x,t) = \int_0^{\infty} a(E)\phi_E(x) e^{-\frac{i}{\hbar}Et}\;dE \;\;\;\;\;\;\;\;\text{\mathrm{(continuous \; spectrum)}}
[/tex]

Quite generally, a wave packet - a superposition of states having different energies - is required in order to have a time-dependence in the probability density and in other observable quantities, such as the average position or momentum of a particle.
Simplest example: a linear combination of just two particular solutions

[tex]\Psi(x,t) = a\phi_E(x) e^{-\frac{i}{\hbar}Et} + b\phi_{E'}(x) e^{-\frac{i}{\hbar}E't}[/tex].

The probability density is given by:

[tex]
|\Psi(x,t)|^2 = |a|^2|\phi_E(x)|^2 + |b|^2|\Psi_{E'}(x)|^2 + 2\mathrm{Re}\left\{a^*b\phi_E^*(x)\phi_{E'}(x)\mathrm{e}^{-i\frac{(E'-E)t}{\hbar}}\right\}
[/tex]

All the time-dependence is contained in the interference term.
 

FAQ: Having trouble with time (in)dependant solutions

1. What are time-dependent solutions in science?

Time-dependent solutions in science refer to mathematical models or equations that describe the behavior or change of a system over time. These solutions take into account the influence of time on the system and are used to predict future outcomes or understand past events.

2. How are time-dependent solutions different from time-independent solutions?

Time-independent solutions do not take into account the influence of time on a system and are typically used to study systems that do not change over time. Time-dependent solutions, on the other hand, are used to study systems that are dynamic and change over time.

3. What are some real-life examples of time-dependent solutions?

Time-dependent solutions are used in many areas of science, including physics, biology, chemistry, and economics. Some examples include predicting the weather, understanding population growth, modeling chemical reactions, and analyzing stock market trends.

4. Why is it important to consider time in scientific solutions?

Time is a crucial factor in understanding and predicting the behavior of systems in the real world. Many natural processes, such as growth, decay, and motion, are time-dependent and cannot be accurately described without considering the influence of time. Therefore, including time in scientific solutions allows for a more comprehensive understanding of the world around us.

5. What are some challenges in finding time-dependent solutions?

One of the main challenges in finding time-dependent solutions is accurately modeling and predicting the behavior of complex systems. Time-dependent solutions often involve multiple variables and can be highly sensitive to initial conditions, making them difficult to solve. Additionally, some systems may have nonlinear behaviors, which can further complicate the search for time-dependent solutions.

Back
Top