Velocity operator inconsistency and discrete particle

[Frostal]

I'm going to mix a couple questions together instead of creating a new topic for each question.
I hope you don't mind.
I'm an electrical engineer(micro-electronics), so while I got the basics of QM in my studies I had to do most of my more 'in depth' learning on my own by reading books/ocwm from MIT.

I took a course in Semiconductor Physics in my last year and came across some questions that seemed to be more of a coincidence rather than a rule. Thus here goes:

1.H$$\Psi = E_{bloch}\Psi$$ for a bloch electron in a crystal lattice.
with $$\Psi$$= u(r,t)e($$\vec{k} \vec{r}$$)

So far nothing out of the ordinary. Now we apply QM's to the wave function to find the average velocity by $$\int \Psi* \nabla_{\vec{r}}\Psi = \nabla_{\vec{k}}E(\vec{k})$$

The last equation is said to be derivated from certain properties. Nothing wrong with that I am sure it's correct. But it's a strange coincidence that this happens to be equal to the group velocity of a packet of Bloch functions.
By assuming the wave packet of the bloch functions and reworking the Hamiltonian we get

$$E_{n}(\nabla_{\vec{r}}) + V(\vec{r})$$
Now by going to the classical limit with $$\nabla_{\vec{r}} = \vec{p}$$ and $$\vec{r} = \vec{r}$$
we get $$E_{n}(\vec{p}) + V(\vec{r})$$

In classical Mechanics we get
$$\vec{v} = \frac{\partial H}{\partial \vec{p}} = \frac{\partial E_{n}(\vec{p})}{ \partial \vec{p}}= \nabla_{\vec{k}} E_{n}(\vec{k})$$
and
$$\vec{F} = \frac{\partial H}{\partial \vec{r}} = \frac{\partial V(\vec{r})}{\partial \vec{r}}$$

it seems that just applying the velocity operator on the Bloch wave or finding the speed through the wave packet give the same speed indications. Which seems weird to me.

I'm also not completely sold on the velocity operator = momentum operator / mass.
Because the equality stems from the free particle where the frequency of the Broglie wave function = $$\frac{p^{2}}{2m}$$. But this is not the case. The Bloch function is made up of several momentum waves(fourier) which propagate with the same speed since the the Bloch wave function acts with frequency $$E_{n}(\vec{k})$$

Only when you use a wavepacket of Bloch waves do you get different Bloch waves move at different speed resulting in a different group velocity from the fase velocity.

I'm very confused in that the velocity operator does not hold into account the time evolution. This is understandable in the case of the free wave particle where the frequency is in direct relation to it's momentum because of the lack of a potential function.
But How can this hold when we get a potential function within the hamiltonian like for example with Bloch electrons. The momentum operator holds but I don't see how the velocity operator could hold in this case.


2.I've gone through a lot of threads on this forum to find a clear cut answer but I doubt there is any so just in case I'm just asking it again here.

As we know there's a lot of debate upon the interpretation of the wave function in QM.
I was just wondering if there's any good theories or proof on the particles existing in discreet states. With that I mean (for example with an electron) when we detect it, do we really detect it as a physical little ball (discreet) or is the detection just a interaction with the particle that forces it to localise into a wavepacket and thus remain nonlocal. In other words is there any proof of the electron and other particles existing outside of these wavefunctions. I know what the wavefunctions mean so don't try to explain the copenhagen interpretation again. I'm just wondering if 'the chance you find the particle in this location' = 'the chance the wave functions crumbles up into a localised wavepacket in with the average position in this location'
Both are very different.

The second interpretation would seem to be the more logical one. Since else particles would teleport at random? Take for example a localised particle in a wavepacket. Now let's say we measure the momentum of the electron. QM states that this particle will collapse into 1 Broglie wavefunction and thus the electron can be anywhere even a lot further into space than it was in the localised wavepacket. Thus the particle would teleport.

I'm a big fan of QM. non locality is the only way for a particle to move in a continuous space since else it would have to pass through an infinite number of 'points' to move. That is if you consider space to be continuous.(I've considered space at a very small level to be discrete but it just doesn't feel right.

 

[azntoon]

To me it feels like space is continuous but that particles can only move in discreet steps)The answer to your first question is that the coincidence you are describing is due to the fact that the Schrödinger equation is a general equation that is valid for both classical and quantum systems. In classical mechanics, the Hamiltonian (H) is a function of momentum (p) and position (r). In quantum mechanics, the Hamiltonian is a function of momentum (k) and position (r). The wave function is a function of both momentum and position. Therefore, when you apply the velocity operator to the wave function, you get a result that is equal to the group velocity of a packet of Bloch functions. This is because the Hamiltonian contains the same information about momentum (in this case k) as the wave function does.As for the second question, there is no definitive proof either way. However, the most accepted theory is that particles exist in a superposition of discrete states until they are observed or otherwise interacted with. This means that particles appear to "teleport" when they are measured or interact with something else, but this phenomenon is actually just the wave function collapsing into a single state.

 

[denian]

I understand your confusion and questions about the velocity operator inconsistency and the nature of particles in quantum mechanics. I will try my best to provide a response that addresses both of your questions.

Firstly, let's address the velocity operator inconsistency. It is important to understand that the velocity operator in quantum mechanics is not the same as the classical velocity in classical mechanics. In quantum mechanics, the velocity operator is defined as the time derivative of the position operator, and it represents the rate of change of the position of a particle. This is different from the classical definition where velocity is the rate of change of position with respect to time.

In the case of Bloch electrons in a crystal lattice, the wave function can be described as a superposition of plane waves with different momenta. When we apply the velocity operator to this wave function, we are essentially taking the derivative with respect to the momentum. This is consistent with the classical definition of velocity, where velocity is the derivative of the momentum with respect to time.

Now, regarding your second question about the nature of particles in quantum mechanics, it is important to note that the wave function is not a physical entity, but rather a mathematical representation of a particle's state. The wave function describes the probability of finding a particle at a certain location, but it does not mean that the particle exists as a physical ball at that location.

The Copenhagen interpretation of quantum mechanics states that the wave function collapses upon measurement, meaning that the particle's state becomes localized at a specific location. However, there are other interpretations of quantum mechanics that suggest that the wave function represents a real physical entity, and the collapse upon measurement is not a fundamental aspect of reality.

Ultimately, the debate about the nature of particles in quantum mechanics is ongoing, and there is no definitive answer. It is up to each individual scientist to interpret the results of experiments and theories in a way that makes the most sense to them. The important thing is to continue exploring and questioning the fundamental principles of quantum mechanics and pushing the boundaries of our understanding.