hokhani said:All right, thanks
But, what is the analytic justification here?
Do you understand how curvature of the wave function connects to its kinetic energy? That's a key point, but it's not the whole story because there are two very different flavors of bound states. One is generally framed in terms of a positive potential energy, and it often extends to infinity (or is even infinite at infinity), as for the particle in a box or the simple harmonic oscillator. This first type generally uses the reference potential to be zero at the center of the spatial domain, and then the lowest number of nodes will have the smallest kinetic energy and the least (positive) total energy. But there is a second type, where the potential energy is usually regarded as negative, and zero at infinity (this is often made necessary by the potential energy going to negative infinity at the center of the spatial domain of interest, as for the Coulomb potential). With this second type, the lower number of nodes actually correspond to higher kinetic energies, because the scale of the wavefunction is so much smaller that it has higher curvature even though it has fewer nodes. But because the potential energy is negative and large at the center, the more concentrated wavefunctions, with fewer nodes and higher kinetic energies, end up having the smallest total energy because the total energy has the largest magnitude but is negative.hokhani said:But i am trying to explain it by using the concept of curvature of the wave function.
The even and odd Eigen states refer to the symmetry properties of quantum mechanical systems. In general, the even Eigen states have a center of symmetry, while the odd Eigen states do not. This difference in symmetry leads to a difference in energy levels, with the even Eigen states having lower energy than the odd ones.
Yes, the difference in energy between even and odd Eigen states is due to the effect of a quantum mechanical phenomenon known as the Pauli exclusion principle. This principle states that two identical fermions (particles with half-integer spin) cannot occupy the same quantum state at the same time. As a result, the energy levels of even Eigen states are lower because they allow for more fermions to occupy them without violating the Pauli exclusion principle.
The difference in energy between even and odd Eigen states plays a crucial role in determining the stability and behavior of quantum systems. Since the even Eigen states have lower energy, they are more likely to be occupied by fermions, making them more stable compared to the odd Eigen states. This difference in stability can affect the way particles interact and behave in a quantum system.
Yes, the concept of even and odd Eigen states has significant implications in various fields, including chemistry and material science. For example, the electronic structure of atoms and molecules can be better understood by considering the difference in energy between even and odd Eigen states. Additionally, the properties of solid materials, such as their conductivity and magnetism, can also be explained using this concept.
Yes, the difference in energy between even and odd Eigen states can be observed experimentally through various techniques such as spectroscopy. By analyzing the energy levels of a quantum system, scientists can determine whether the system shows a preference for even or odd Eigen states. This experimental evidence further supports the concept of symmetry and its role in determining the energy levels of quantum systems.