Does a Free Particle in Quantum Mechanics Have Constant Energy?

[xcorat]

Hi, I'm new to QM (and phy forum :D), and studyin alone..!

so, if you consider the free particle, it does not have a solution in the form of separable solutions(??). Which means that if the same experiment (independent, of course,) is carried out total energy is different each time at t = t_a (for constant a).

what does this mean? it seems to me like energy is created and destroyed. Is it so? or does it have to do som'n with uncertainness principle?

help me, thax a lot.

 

[muppet]

Hi xcorat, and welcome to PF
The schroedinger equation for a free particle DOES admit a separable solution- why do you think it doesn't? The general condition for this to be the case is that the potential term in the hamiltonian doesn't depend on time: V(x,t) =V(x). Here, V=0 ...
As for energy being created and destroyed... it's more true to say that the system doesn't have an energy- at least, not to begin with. The way it works is this: solving the Schroedinger equation for a time-independent potential gives you the "spectrum" of the hamilton- the range of possible energy eigenvalues. A completely free particle- which has never interacted with anything, ever- can theoretically attain any of these values. Then, when you measure its energy, its energy assumes a definite value, which it keeps for ever. It's not a physically realistic picture- it's just a very easy differential equation to solve. Realistic descriptions usually involve harder maths! There's been millions of discussions of the uncertainty principle on here before- search the forums, read a few, then come back if you get stuck.

 

[Entropia]

Hello and welcome to the world of quantum mechanics! It's great to see new people interested in this fascinating field of study.

To answer your question, the free particle is a concept in quantum mechanics that refers to a particle that is not subjected to any external forces or interactions. In other words, it is not bound to any potential energy well or barrier. In classical mechanics, this would mean that the particle would continue to move with a constant velocity forever.

However, in quantum mechanics, things are a bit different. The wave function of a free particle is described by a plane wave, which does not have a definite energy associated with it. This means that the energy of the particle is not fixed, but rather it can take on a range of values. This is where the uncertainty principle comes into play.

The uncertainty principle states that there is a limit to how precisely we can measure certain pairs of physical properties, such as position and momentum. In the case of a free particle, the more precisely we know the position of the particle, the less we know about its momentum, and vice versa. This leads to a range of possible energies for the particle.

So, to answer your question, the energy of a free particle is not created or destroyed, but rather it is uncertain due to the nature of quantum mechanics. This is one of the fundamental principles of quantum mechanics that sets it apart from classical mechanics.

I hope this helps to clarify things for you. Keep studying and exploring the world of quantum mechanics, and don't hesitate to reach out with any further questions. Good luck!

 

[Entropia]

Hi there! Welcome to the world of quantum mechanics. It's great that you're studying on your own, but I would highly recommend seeking out some resources to help guide your learning.

To answer your question, the concept of a free particle in quantum mechanics is a bit different from what we are used to in classical mechanics. In classical mechanics, a free particle has a well-defined position and momentum, and its energy is constant. However, in quantum mechanics, a free particle does not have a well-defined position or momentum.

This is because in quantum mechanics, particles behave like waves, and their position and momentum are described by wavefunctions. The wavefunction of a free particle is a plane wave, which means it extends infinitely in space. Therefore, the position and momentum of a free particle are not well-defined, and its energy can vary.

This does not mean that energy is created or destroyed. Instead, it is a result of the uncertainty principle, which states that we cannot know both the position and momentum of a particle with absolute certainty. The more we know about one, the less we know about the other.

I hope this helps clarify things for you. Keep studying and exploring the fascinating world of quantum mechanics! Best of luck in your studies.