Is Total Energy Always Double the Kinetic Energy in Quantum Mechanics?

[bon]

Homework Statement

Ok so (1) E = hf and (2) lambda = h/p

Homework Equations

The Attempt at a Solution

For a particle mass m, speed v, momentum p

Surely if p^2 / 2m = Ek (kinetic energy)

then we can write from (2) Ek = h^2/ 2m lambda^2

But from (1) we can write E = hv/lambda => E = h(p/m)/lambda => E = h^2 / m lambda^2

This seems to imply that for such a particle total energy is always double KE. What's gone wrong?

 

[Delphi51]

p^2 / 2m = Ek applies to ordinary particles, not photons.
Photons are WEIRD. The fact that p = h/λ applies to both is incredible and I think de Broglie thought of it while drinking beer.

 

[bon]

Yes but I am talking about an ordinary particle here..
Does E=hf not work for ordinary particles?

 

[Delphi51]

E = hf applies to photons only.

 

[rwisz]

It seems like there may be a mathematical error in your attempt at a solution. Let's take a closer look at the equations and see if we can figure out where the confusion lies.

Equation (1) states that the energy of a particle is equal to Planck's constant (h) multiplied by its frequency (f). This equation is typically used to describe the energy of a photon, which is a massless particle. For a particle with mass, the energy equation would be E = hf + KE, where KE is the kinetic energy.

Equation (2) relates the particle's wavelength (lambda) to its momentum (p) through Planck's constant. This equation is often used to describe the behavior of particles with mass, as it takes into account both their wavelength and momentum.

In your attempt at a solution, you seem to have mixed up these equations and tried to use them interchangeably. This is where the apparent contradiction arises. The equations are meant to be used separately, depending on the type of particle you are studying.

Additionally, the equation E = p^2 / 2m is the classical equation for kinetic energy, which may not accurately describe the behavior of particles at the quantum level. It is important to use the correct equations and understand their limitations when studying particles and their energy.

In summary, the apparent contradiction is likely due to a misunderstanding or misuse of the equations. It is important to carefully consider the equations and their applicability to the situation at hand in order to avoid such contradictions.