Wave function using the time dependent Schrodinger equation

In summary, the time-dependent Schrödinger equation can be solved using the formal solution, which involves the propagator ##G(x,x',t)##. This can be calculated by using the momentum eigenstates for a free particle.
  • #1
ThiagoSantos
2
1
Given a wavefunction ψ(x, 0) of a free particle at initial time t=0, I need to write the general expression of the function at time t. I used a Fourier transform of ψ(x, t) in terms of ψ(p, t), but, i don't understand how to use green's functions and the time dependent schrodinger equation to get my answer. What's the relationtship between them?
 
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  • #2
The time-dependent Schrödinger equation reads
$$\mathrm{i} \hbar \partial_t \psi(x,t)=\hat{H} \psi(x,t).$$
Let's assume for simplicity that
$$\hat{H}=\frac{\hat{p}^2}{2m} + V(\hat{x}),$$
i.e., that ##\hat{H}## is not explicitly time dependent. Then the formal solution of the equation above is
$$\psi(x,t)=\exp \left (-\frac{\mathrm{i} \hat{H} t}{\hbar} \right) \psi(x,0).$$
This you can write in the form
$$\psi(x,t)=\int_{\mathbb{R}} \mathrm{d} x' \left \langle x \left |\exp \left (-\frac{\mathrm{i} \hat{H} t}{\hbar} \right) \right| x' \right \rangle \psi(x',0)= \int_{\mathbb{R}} \mathrm{d} x' G(x,x',t) \psi(x',0),$$
i.e., the propator is
$$G(x,x',t)=\left \langle x \left |\exp \left (-\frac{\mathrm{i} \hat{H} t}{\hbar} \right) \right| x' \right \rangle.$$
Usually it's of course difficult to really calculate the propagator.

For a free particle, where ##\hat{H}=\hat{p}^2/(2m)## you can use the momentum eigenstates to evaluate it:
$$G(x,x',t)=\int_{\mathbb{R}} \mathrm{d} p \langle x |\exp[-\mathrm{i} \hat{p}^2 t/(2m \hbar)]|p \rangle \langle p|x' \rangle.$$
 

1. What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wave function of a quantum system changes over time.

2. What is the wave function?

The wave function is a mathematical representation of the state of a quantum system. It contains all the information about the system's position, momentum, and other physical properties.

3. How is the time dependent Schrödinger equation used?

The time dependent Schrödinger equation is used to calculate the evolution of a quantum system over time. It allows us to determine the probability of finding the system in a particular state at a given time.

4. What is the difference between the time dependent and time independent Schrödinger equation?

The time dependent Schrödinger equation takes into account the time evolution of a quantum system, while the time independent Schrödinger equation only describes the stationary states of the system.

5. What are the applications of the time dependent Schrödinger equation?

The time dependent Schrödinger equation is used in a wide range of applications, including quantum chemistry, solid state physics, and quantum computing. It is also essential for understanding the behavior of particles at the atomic and subatomic level.

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