Energy eigenvalue and mass inverse relation?

[AbbasB.]

So, after time-independent 1D Schrödinger equation is solved, this is obtained

E = n²π²ħ²/(2mL²)

This means that the mass of the 'particle' is inversely related to the energy eigenvalue.
Does this mean that the actual energy of the particle is inversely related to its mass?
Isn't this counter intuitive? Doesn't E = mc²?

Put in another way, what does E mean in the first equation? Is the eigenvalue of energy different than our classical notion of energy?

 

[DrClaude]

This is non-relativistic quantum mechanics, so the energy doesn't include mc² or a correction due to relativistic momentum. m is simply the mass of the system ("rest mass"). It is a fixed property of the system, and doesn't depend on the actual energy.

 

[Orodruin]

You are mixing special relativity and quantum mechanics in a way which is not compatible. What you should be doing is to compare the energy and momentum with the classical expressions where $E = p^2/2m$. The quantised energy levels have $p^2 = n^2 \pi^2 \hbar^2 /L^2$. In addition, we are here considering only kinetic energy. The classical expression also holds for the kinetic energy in relativity when the momentum is small: $E_k = \sqrt{m^2 c^4 + p^2 c^2} - mc^2 \simeq p^2/2m$.

 

[AbbasB.]

My doubt is simply the following (discounting the idea E = mc^2):
The m in the equation is the mass of the 'particle', how is it inversely related to the energy? What does the equation even mean? Also, I must add, the equation was derived when TISE was solved for a box of 1D, that is, the particle was bound in a potential well.

 

[Orodruin]

My doubt is simply the following (discounting the idea E = mc^2):
The m in the equation is the mass of the 'particle', how is it inversely related to the energy? What does the equation even mean? Also, I must add, the equation was derived when TISE was solved for a box of 1D, that is, the particle was bound in a potential well.

You are thinking about it in the wrong way. It is the energy of the Hamiltonian eigenstates which are inversely proportional to the mass $m$ - it is not the mass which depends on $E$. The mass is a fixed parameter in QM and your solution for the quantised energy levels depends on it.

 

[AbbasB.]

You are thinking about it in the wrong way. It is the energy of the Hamiltonian eigenstates which are inversely proportional to the mass $m$ - it is not the mass which depends on $E$. The mass is a fixed parameter in QM and your solution for the quantised energy levels depends on it.

Accepted. Okay. So, say, I come to the relation, what does it mean now? Okay, the mass is constant. But, here's the problem:
Consider two masses, one of an electron, the other of a ball of mass 1 kg.

Plug both in the same equation. I will get the value of E to be greater for an electron, and less for the ball. What will that mean? The ball obviously has more energy than the electron, then why this inverse connection? Am I still looking it in the wrong way, in that, is my conception of a wave function flawed? Both will have different wave functions, but how will they differ? Can you define a wave function for a ball (since it is a wave packet)?

 

[Orodruin]

Accepted. Okay. So, say, I come to the relation, what does it mean now? Okay, the mass is constant. But, here's the problem:
Consider two masses, one of an electron, the other of a ball of mass 1 kg.

Plug both in the same equation. I will get the value of E to be greater for an electron, and less for the ball. What will that mean? The ball obviously has more energy than the electron, then why this inverse connection? Am I still looking it in the wrong way, in that, is my conception of a wave function flawed? Both will have different wave functions, but how will they differ? Can you define a wave function for a ball (since it is a wave packet)?

It means nothing because you cannot constrain the ball to be in the same type of microscopic box as an electron. Also note that these are the energy eigenstates of the particle - it only tells you which energy values are allowed for the particle. For a larger mass, it only means that the distance between energy levels is smaller because the same change in momentum results in a lower change in the energy for a heavy particle.

I have also changed the thread level to "I". Note that labelling a thread "A" means that you would like the discussion to be at the level understandable by a graduate student in physics.

 

[AbbasB.]

Thank you.