Equation of energy of a particle

[justwild]

Suppose a particle moves along Cartesian coordinates from x=0 to x=a, such that it can move in definite regions having definite energy which corresponds to formation of standing waves with nodes at its ends, that is, x=0 and x=a. Let the particle be of mass m and moving with velocity v.
Moreover, let n denotes the number of loops in the standing wave, which can take values 1,2,3...
...
Now I am deciding to derive the energy associated with the particle...
.........
for standing waves having ends with nodes, we have
a=$nλ/2$
$\Rightarrow$ λ=$2a/n$
$\Rightarrow$ $\nu$=v/λ
$\Rightarrow$ E=h$\nu$=nv/2a

Now the particle has linear momentum as p=mv$\Rightarrow$v=p/m , where p is the linear momentum which remains constant for a given energy level (I am using this as an analogy to angular momentum as per Bohr's postulates)

substituting in equation of energy we have,
E=np/2am

Now I have two questions...the analogy I have used for momentum...was that correct?
and E=h$\nu$ is applicable for EM waves which has speed of light...Is this applicable if particle's speed is not equal to speed of light?

 

[eljose79]

I would like to address your questions and provide some clarification. Firstly, the analogy you have used for momentum is not entirely correct. In classical mechanics, linear momentum is defined as p=mv, where m is the mass of the particle and v is its velocity. In quantum mechanics, momentum is described by the operator \hat{p}=-i\hbar\frac{\partial}{\partial x}, which is different from the classical definition. However, in certain situations, such as in the case of a free particle, the classical and quantum descriptions of momentum are equivalent. In your case, since the particle is moving with a constant velocity, the classical definition of momentum can be used.

Secondly, the equation E=h\nu is applicable for all types of waves, not just electromagnetic waves. It is a fundamental equation in quantum mechanics known as the Planck-Einstein relation, where h is the Planck constant, \nu is the frequency of the wave, and E is the energy of the particle. This equation applies to all types of waves, including matter waves, which are associated with particles.

In summary, the analogy you have used for momentum is not entirely correct, but it can be used in certain situations. The equation E=h\nu is applicable for all types of waves, including matter waves. I hope this helps clarify your questions. Keep up the good work in your studies of quantum mechanics!