Sorry, I can't understand your statement. Maybe you have strayed from the point.azntoon said:The idea here is that when the momentum operator p is applied to an eigenstate, it will produce a state with the same energy (eigenvalue) as before. However, when the momentum operator squared, p^2, is applied to this same eigenstate, the result will be a state with a different energy. This is because the momentum operator squared contains additional terms corresponding to higher powers of momentum, which require higher energies to produce states with the same eigenvalue. This is an example of what is known as "quantum tunneling", where particles can pass through "barriers" of energy which would normally be too high to be overcome. In this case, the particle is able to "tunnel" through the barrier by utilizing the energy associated with its momentum.
Where? Please give a reference.Jiangwei Du said:Someone says
You can establish a linear dispersion relation with a term like ##v \mathbf{\sigma} \cdot \mathbf{p}## and you can add it your p^2 term to get some generalised k.p approximation useful for some semiconductors/semimentals. Is this what is motivating your question?Jiangwei Du said:TL;DR Summary: We have already know the energy eigenvalue E0 of initial Hamiltonian H0. So when we add the extra item-λp/m, how the energy eigenvalue will vary?
Someone says we can choose the new eigenstate: exp(-iλx/hbar)*ψ,and let the momentum operator p acts upon this new state. At the same time, so does p^2. Something miraculous will happen afterwards. My question is: how to image this point? Thank you very much.
To calculate the energy eigenvalue of the Hamiltonian, you need to solve the Schrödinger equation, which is a differential equation that describes the behavior of quantum systems. This equation involves the Hamiltonian operator, which represents the total energy of the system. By solving the Schrödinger equation, you can obtain the energy eigenvalues of the system.
The Hamiltonian operator is a mathematical operator that represents the total energy of a quantum system. It is composed of two parts: the kinetic energy operator (H0), which represents the energy of the system due to its motion, and the potential energy operator (λp/m), which represents the energy of the system due to its interactions with its surroundings.
The Hamiltonian operator plays a crucial role in determining the energy eigenvalues of a quantum system. It is used in the Schrödinger equation to describe the behavior of the system and its interactions with its surroundings. The energy eigenvalues are the possible values of the total energy of the system, which is represented by the Hamiltonian operator.
The energy eigenvalues of a quantum system represent the possible states that the system can exist in. Each energy eigenvalue corresponds to a specific energy level of the system, and the probability of the system being in that state is given by the square of the wave function. The energy eigenvalues also determine the energy spectrum of the system, which is important in understanding the behavior and properties of the system.
Yes, there are several techniques for finding the energy eigenvalues of the Hamiltonian. One common method is to use numerical methods, such as the finite difference method or the variational method, to approximate the solutions of the Schrödinger equation. Another approach is to use analytical methods, such as perturbation theory or the variational principle, to obtain approximate solutions. The choice of method depends on the complexity of the system and the accuracy required for the energy eigenvalues.