Originally posted by Sacroiliac
In the hydrogen atom the 1s orbital has a clearly defined energy of 13.6 eV, but the probability density and radial probability density says you are liable to find the 1s electron anywhere from the nucleus on out.
How does this exact energy value jive with this variable position?
Originally posted by Tyger
can be exact with the position not being exact because the canonical conjugate to the the position, the momentum, also varies. To put it very crudely, the variation in the momentum compensates for the variation in position, allowing the energy to have an exact value.
Originally posted by Sacroiliac
Are you saying that as the particle gets closer to the nucleus its potential energy decreases but because it is more localized the HUP causes its momentum to increase and exactly offset the decrease in Potential energy?
In quantum mechanics, energy is a fundamental quantity that describes the state of a system. It can exist in discrete amounts, known as energy levels, and can be transferred between particles or converted into different forms. Energy in quantum mechanics is described by the Hamiltonian operator, which is used to calculate the total energy of a system.
The Heisenberg uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. This also applies to energy and position in quantum mechanics, as the more precisely we measure the position of a particle, the less we know about its energy and vice versa. This principle is a fundamental aspect of quantum mechanics and is related to the probabilistic nature of particles at the quantum level.
In classical mechanics, energy is considered to be continuous and can take on any value. However, in quantum mechanics, energy is quantized, meaning it can only exist in discrete amounts or levels. This is due to the wave-particle duality of particles at the quantum level and is a key difference between classical and quantum mechanics.
In quantum mechanics, the position of a particle is described by a wave function, which represents the probability of finding the particle at a certain position. The position of a particle can affect its energy, as the wave function and energy are related through the Schrödinger equation. Changes in the position of a particle can lead to changes in its energy level, which can be observed through phenomena such as quantization and tunneling.
In quantum mechanics, potential energy is a form of energy that is associated with the position of a particle within a potential field. This potential energy can affect the total energy of a system and is related to the Hamiltonian operator. By solving the Schrödinger equation, we can determine the energy levels and potential energy of a particle in a given system, providing insight into its behavior and properties.