Question

Consider an electron in the ground state of a hydrogen atom. (a) Calculate the expectation value of its potential energy. (b) What is the expectation value of its kinetic energy? (Hint: What is the expectation value of the total energy?

Short answer

The expectation value of the potential energy is \(-27.2\) eV. The expectation value of the kinetic energy is \(27.2\) eV.

Step-by-step solution

Understanding the ground state energy

The ground state energy \(E\) of a hydrogen atom is \(-13.6\) eV.

Energy conservation

The total energy is the sum of kinetic and potential energies. Since we know the energy \(E\) is \(-13.6\) eV, and the potential energy is always twice the total energy in the hydrogen atom ground state, then the potential energy \(U\) is \(U = 2E = -27.2\) eV.

Calculate expectation value of kinetic energy

With the same reasoning used in the previous step, considering the energy conservation equation, we can deduce that the kinetic energy is opposite to potential energy in a hydrogen atom ground state: thus, the expectation value of kinetic energy \(K = -U = 27.2\) eV.

Key concepts

Expectation Value of Energy

In quantum mechanics, the expectation value is a form of an average. More specifically, for an electron in a hydrogen atom, the expectation value represents the average energy that we would observe if we measured the energy of the electron many times. This concept is crucial because unlike in classical physics, quantum mechanics tells us that we cannot know the exact energy of a particle at a given moment but can predict the most likely outcomes.
When dealing with an electron in the ground state of a hydrogen atom, we apply the quantum mechanical concept to determine the expectation value of both the potential and kinetic energies. Here, the ground state refers to the lowest energy state possible for the electron, and the expectation values are derived based on the wave functions that describe the electron’s behavior. For the ground state, the expectation value of the potential energy has been found to be negative, which indicates that we need to supply energy to the electron to remove it from the atom's influence.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at very small scales, such as electrons in atoms. At this quantum level, particles don't follow the same rules as objects we're familiar with in our day-to-day lives, hence the notion of a particle with a well-defined position and velocity is replaced by wave functions and probabilities.
The wave function contains all the information about a particle, and the probability of finding an electron in a particular region is determined by the square of the wave function's amplitude. This probabilistic framework governs the behavior and interactions of particles, leading to the fascinating and often counterintuitive observations of the quantum world, like the discrete energy levels observed in the hydrogen atom.

Energy Conservation in Quantum Systems

The principle of energy conservation holds true even in the quantum world: the total energy of a closed system remains constant over time. In the context of the hydrogen atom, this total energy is made up of kinetic and potential energy.
However, in quantum mechanics, these energies are given by the expectation values, not by single, definite numbers. This is because the electron is not behaving like a billiard ball moving in a clear trajectory, but rather as a wave of probabilities. Despite the probabilistic nature, the expectation values still adhere to energy conservation laws, meaning the total energy in the system, which is a sum of the kinetic and potential energy expectation values, remains constant within an isolated quantum system.

Potential and Kinetic Energy in Atoms

Within an atom, electrons possess both kinetic and potential energy. The kinetic energy arises from the movement of the electron, while the potential energy stems from the electrostatic force between the positively charged nucleus and the negatively charged electron.
In the special case of a hydrogen atom in its ground state, there is an interesting relationship: the potential energy is always two times the total energy, and the kinetic energy is equal in magnitude but opposite in sign to the potential energy. This is a direct result of the virial theorem, which relates the average total energy of a bound system to its potential energy. These values are expectation values, reflecting the quantum mechanical nature of the electron: they represent average measurements over time of an electron's energies, rather than definitive one-time measurements.