QM, central potential, system collapse

• WarnK
How does that reduce the energy per particle?In summary, the conversation is about a problem involving a homogeneous system of spin 1/2 particles with a potential V. The expectation value of the Hamiltonian in the non-interacting ground state is given by E^(0) + E^(1), where E^(0) is the kinetic energy and E^(1) is the interaction energy. The problem is to prove that the system will collapse, assuming that V is central and spin-independent and that the integral of V(x) is finite. The hint is to consider (E^(0) + E^(1))/N as a function of density, and to show that if the particle number increases, the energy per particle decreases. However,
WarnK

Homework Equations

(this is ~Fetter & Walecka Quantum theory of many-particle systems problem 1.2b)
Homogeneous system of spin 1/2 particles, potential V.
Expectation value of Hamiltonian in the non interacting ground state is
$$E^{(0)} + E^{(1)} = 2 \sum_k^{k_F} \frac{\hbar^2 k^2}{2m} + 1/2 \sum_{k \lambda}^{k_F} \sum_{k' \lambda'}^{k_F} \big[ <k\lambda k'\lambda'|V|k\lambda k'\lambda'> - <k\lambda k'\lambda'|V|k'\lambda' k\lambda>\big]$$

Assume V is central and spin independant
V(|x_1-x_2|) < 0 for all |x_1-x_2|
The intergral of |V(x)| is finite

Prove that the system will collapse.
Hint: start from $$( E^{(0)} + E^{(1)} ) / N$$ as a function of density

The Attempt at a Solution

If the energy per particle, $$( E^{(0)} + E^{(1)} ) / N$$, is reduced when the density increases (which I guess means that N increases) the system will collapse.
But how do I show that, I don't see what I can do.

If the particle number increases, the sums might have a few more terms, how does that reduce the number of particles?

What is quantum mechanics?

Quantum mechanics is a branch of physics that studies the behavior and interactions of particles on a microscopic scale. It explains the behavior of matter and energy at the subatomic level and is essential for understanding the fundamental laws of the universe.

What is a central potential in quantum mechanics?

A central potential is a type of potential energy that depends only on the distance between two particles and not on their orientation. In quantum mechanics, this type of potential is often represented by the Coulomb potential, which describes the interaction between charged particles.

What is the system collapse in quantum mechanics?

System collapse, also known as wave function collapse, is a concept in quantum mechanics that explains the sudden change or collapse of a particle's wave function when it is observed or measured. This collapse is a fundamental aspect of quantum mechanics and is necessary for understanding the probabilistic nature of particles at the subatomic level.

How does the uncertainty principle relate to quantum mechanics?

The uncertainty principle, also known as Heisenberg's uncertainty principle, is a fundamental concept in quantum mechanics that states that it is impossible to know both the position and momentum of a particle with absolute certainty. This principle highlights the probabilistic nature of particles in the quantum world and is essential for understanding the limitations of our knowledge at the subatomic level.

What are the applications of quantum mechanics?

Quantum mechanics has many practical applications, including in the development of new technologies such as transistors, lasers, and computer memory. It also plays a critical role in fields such as chemistry, materials science, and nanotechnology. Additionally, quantum mechanics has led to groundbreaking discoveries in areas such as quantum computing, cryptography, and quantum teleportation.

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