General solution of the Schrodinger equation for a free particle?

[21joanna12]

Homework Statement

I'm trying to figure out how the general solution of the Schrodinger equation for a free particle when v=0 relates to anything I have learned in class...

Homework Equations

For Eψ=(hbar²/2m)d²ψ/dx₂

The Attempt at a Solution

I really have no idea- what is confusing me is that ψ is a function rather than a constant so I don't know how to treat it...

Thank you all for any help!

And I really apologise for the equation... I don't really understand how to use LaTeX

 

[ehild]

Homework Statement

I'm trying to figure out how the general solution of the Schrodinger equation for a free particle when v=0 relates to anything I have learned in class...

Homework Equations

For [STRIKE]Eψ=(hbar²/2m)d²ψ/dx₂
[/STRIKE]

You miss the negative the sign: $-\hbar^2/(2m)d^2ψ/dx^2=Eψ$

That is a differential equation of form y"+kx=0. What is the solution?

The former equation is the time independent equation for one dimension. The time dependent equation is

$i\hbar \frac{\partial ψ}{\partial t}=-\hbar^2/(2m)\frac{\partial^2}{\partial x^2}$

ehild

 

[haruspex]

That is a differential equation of form y"+kx=0.

y"+ky=0?

 

[ehild]

I meant it so. Thanks.

ehild

 

[HalfLostAndSearching]

I can understand your confusion about the general solution of the Schrodinger equation for a free particle. It is a complex and abstract concept that may be difficult to grasp at first. However, it is an important concept in quantum mechanics and has many applications in understanding the behavior of particles at the microscopic level.

To provide some context, the Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system over time. It is used to determine the probability of finding a particle at a particular location and time. In the equation, ψ represents the wave function of the particle, which is a mathematical function that describes the quantum state of the particle. The term v represents the potential energy of the particle, which is zero for a free particle.

Now, to understand the general solution of the Schrodinger equation for a free particle, we can start by looking at the equation itself. As you have correctly stated, it is an eigenvalue equation where the energy of the particle (E) is related to the wave function (ψ) through the second derivative with respect to position (x).

The general solution for this equation is a wave function that takes the form of a plane wave, which has the form e^(ikx), where k is the wave vector. This means that the wave function of a free particle is a traveling wave, with a constant velocity (v) determined by the wave vector. This is why it is called a "free" particle since it is not confined or affected by any potential energy.

In terms of what you have learned in class, this solution may relate to other concepts such as wave-particle duality, the uncertainty principle, and the quantization of energy levels. It is also important to note that this is a simplified solution for a free particle and may not fully capture the behavior of more complex quantum systems.

I hope this explanation has helped to clarify the general solution of the Schrodinger equation for a free particle. Keep in mind that this is just one aspect of a vast and complex field, and there is always more to learn and understand. Keep exploring and asking questions, and you will continue to deepen your understanding of quantum mechanics.