# What is the “lowest energy”?

In many textbooks I come across the term lowest energy. For example in atomic structures, electrons are placed in orbitals in order for the atom to have the lowest energy. But what is this energy? Potential- or kinetic energy or the sum of the two?

• The sum of the two, usually the eigenvalue of the Hamiltonian operator. – Lewis Miller Dec 15 '18 at 23:21

The lowest energy of a quantum system is the minimum eigenvalue of the Hamiltonian of the system. The Hamiltonian is the operator which corresponds to the total energy of the system, so it is the sum of the kinetic and potential energy. This is often also referred to as the ground state energy of the system.

but what is this energy?

Heuristically, the ground state energy is the energy which the system has simply by existing. Normally, this doesn't make any sense: If I just make the system do nothing, just sit there, then it would have zero energy. We wouldn't need the fancy name.

But it turns out that this is impossible. We can never "pin down" a system and make it completely still, due to the famous Heisenberg uncertainty principle, $$\Delta x \Delta p \geq \hbar/2.$$ The system must have some momentum, to satisfy the inequality. In turn, it will also have some energy. This is why we can never reach absolute zero! No matter how hard we try, there always remains the jiggling of the ground state.

• Writing that it is not possible to get absolute zero because the system has to satisfy the uncertainty principle is completely wrong. Absolute zero does correspond to the minumum of energy whatever it is. It has nothing to do with energy being zero. One very good reason is that the energy is always definite within an arbitrary constant. – GiorgioP Dec 16 '18 at 8:48
• @GiorgioP Thank you for pointing out my mistake. Unfortunately, I always thought that this was the case, and I don't understand why it shouldn't be. Is it completely wrong, absolutely and utterly? Could you enlighten me? – Hanting Zhang Dec 16 '18 at 16:42
• Think of a particle of mass $m$ in a potential well which could be approximated, close to the minimum, by a harmonic potential $\frac{1}{2}( m \omega^2 x^2 - \hbar \omega)$. By construction its lowest eigenvalue is zero. The corresponding eigenstate is a minimum uncertainty wavefunction . Moreover, since the system is in an eigenstate, the variance of the energy vanishes, Therefore, all the measurements of energy will return zero. The argument you are citing is usually used to justify a value of the minimum energy higher than the minimum of the potential energy. – GiorgioP Dec 16 '18 at 17:29
• I am using it to justify exactly that. My intent was to make it clear that there is a difference between the classical minimum and the quantum minimum, which is why we give it a name. Also, I still don't understand why the uncertainty principle doesn't prevent a system from reaching absolute zero. Wikipedia seems to agree with me. – Hanting Zhang Dec 16 '18 at 18:04
• What wikipedia says is that the uncertainty principle may help to understand why, when the system gets the minimum of energy there is still some irreducible kinetic energy. But this has nothing to do with absolute zero. Things go the other way round: at absolute zero the system has to get the minimum energy. Due to quantum mechanics such minimum energy state is not the state with zero kinetic energy. what is wrong is to add the implication no zero energy -> no zero temperature. The thermodynamic temperature is not proportional to the kinetic energy at low temperatures. – GiorgioP Dec 16 '18 at 19:09

The sum of the two.

An electronic state like an orbital is an exact or approximate solution of some time independent Schrödinger equation, i.e. an eigenstate of the hamiltonian made by a kinetic and a potential energy term. The corresponding eigenvalue is the expectation value of such hamiltonian, evaluated on the state described by the orbital.

The expectation value of the hamiltonian can always be written as the sum of the expectation value of kinetic and potential energy. The lowest energy, is the eigenvalue of the hamiltonian with the minimum value.

In QM the Schrödinger Equation gives you the solutions for the wavefunction of a particle with a given potential. Because the energy is quantized you usually find several possible values for the energy that are given by an integer number $$n$$ called the Principal Quantum Number. The lowest value of $$E_n$$, normally when $$n=0$$ or $$n=1$$, is the Groud-State. This means that the energy of the particle is the lowest and it is a combination of the potential energy and the kinetic energy.