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.(adsbygoogle = window.adsbygoogle || []).push({});

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....

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for standing waves having ends with nodes, we have

a=[itex]nλ/2[/itex]

[itex]\Rightarrow[/itex] λ=[itex]2a/n[/itex]

[itex]\Rightarrow[/itex] [itex]\nu[/itex]=v/λ

[itex]\Rightarrow[/itex] E=h[itex]\nu[/itex]=nv/2a

Now the particle has linear momentum as p=mv[itex]\Rightarrow[/itex]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[itex]\nu[/itex] is applicable for EM waves which has speed of light...Is this applicable if particle's speed is not equal to speed of light???

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# Equation of energy of a particle

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