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Energy levels for an electron in an atom: ground state and excited states. After absorbing energy, an electron may jump from the ground state to a higher-energy excited state.

The ground state of a quantum-mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In the quantum field theory, the ground state is usually called the vacuum state or the vacuum.

If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator that acts non-trivially on a ground state and commutes with the Hamiltonian of the system.

According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature.

Ground state has no nodes in one-dimension

In one dimension, the ground state of the Schrödinger equation can be proven to have no nodes.^[1]

Consider the average energy of a state with a node at x = 0; i.e., ψ(0) = 0. The average energy in this state would be

⟨ψ|H|ψ⟩=∫dx(−ℏ22mψ∗d2ψdx2+V(x)|ψ(x)|2),{displaystyle langle psi |H|psi rangle =int dx,left(-{frac {hbar ^{2}}{2m}}psi ^{*}{frac {d^{2}psi }{dx^{2}}}+V(x)|psi (x)|^{2}right),}

where V(x) is the potential.

Now, consider a small interval around x=0{displaystyle x=0}; i.e., x∈[−ϵ,ϵ]{displaystyle xin [-epsilon ,epsilon ]}. Take a new (deformed) wave function ψ'(x) to be defined as ψ′(x)=ψ(x){displaystyle psi '(x)=psi (x)}, for x<−ϵ{displaystyle x<-epsilon }; and ψ′(x)=−ψ(x){displaystyle psi '(x)=-psi (x)}, for x>ϵ{displaystyle x>epsilon }; and constant for x∈[−ϵ,ϵ]{displaystyle xin [-epsilon ,epsilon ]}. If ϵ{displaystyle epsilon } is small enough, this is always possible to do, so that ψ'(x) is continuous.

Assuming ψ(x)≈−cx{displaystyle psi (x)approx -cx} around x=0{displaystyle x=0}, one may write

ψ′(x)=N{|ψ(x)|,|x|>ϵ,cϵ,|x|≤ϵ,{displaystyle psi '(x)=N{begin{cases}|psi (x)|,&|x|>epsilon ,\cepsilon ,&|x|leq epsilon ,end{cases}}}

where N=11+43|c|2ϵ3{displaystyle N={frac {1}{sqrt {1+{frac {4}{3}}|c|^{2}epsilon ^{3}}}}} is the norm.

Note that the kinetic-energy density |dψ′dx|2<|dψdx|2{displaystyle left|{frac {dpsi '}{dx}}right|^{2}<left|{frac {dpsi }{dx}}right|^{2}} everywhere because of the normalization. More significantly, the average kinetic energy is lowered by O(ϵ){displaystyle O(epsilon )} by the deformation to ψ'.

Now, consider the potential energy. For definiteness, let us choose V(x)≥0{displaystyle V(x)geq 0}. Then it is clear that, outside the interval x∈[−ϵ,ϵ]{displaystyle xin [-epsilon ,epsilon ]}, the potential energy density is smaller for the ψ' because |ψ′|<|ψ|{displaystyle |psi '|<|psi |} there.

On the other hand, in the interval x∈[−ϵ,ϵ]{displaystyle xin [-epsilon ,epsilon ]} we have

Vavgϵ′=∫−ϵϵdxV(x)|ψ′|2=ϵ2|c|21+43|c|2ϵ3∫−ϵϵdxV(x)≃2ϵ3|c|2V(0)+⋯,{displaystyle {V_{text{avg}}^{epsilon }}'=int _{-epsilon }^{epsilon }dx,V(x)|psi '|^{2}={frac {epsilon ^{2}|c|^{2}}{1+{frac {4}{3}}|c|^{2}epsilon ^{3}}}int _{-epsilon }^{epsilon }dx,V(x)simeq 2epsilon ^{3}|c|^{2}V(0)+cdots ,}

which holds to order ϵ3{displaystyle epsilon ^{3}}.

However, the contribution to the potential energy from this region for the state ψ with a node is

Vavgϵ=∫−ϵϵdxV(x)|ψ|2=|c|2∫−ϵϵdxx2V(x)≃23ϵ3|c|2V(0)+⋯,{displaystyle V_{text{avg}}^{epsilon }=int _{-epsilon }^{epsilon }dx,V(x)|psi |^{2}=|c|^{2}int _{-epsilon }^{epsilon }dx,x^{2}V(x)simeq {frac {2}{3}}epsilon ^{3}|c|^{2}V(0)+cdots ,}

lower, but still of the same lower order O(ϵ3){displaystyle O(epsilon ^{3})} as for the deformed state ψ', and subdominant to the lowering of the average kinetic energy.
Therefore, the potential energy is unchanged up to order ϵ2{displaystyle epsilon ^{2}}, if we deform the state ψ{displaystyle psi } with a node into a state ψ' without a node, and the change can be ignored.

We can therefore remove all nodes and reduce the energy by O(ϵ){displaystyle O(epsilon )}, which implies that ψ' cannot be the ground state. Thus the ground-state wave function cannot have a node. This completes the proof. (The average energy may then be further lowered by eliminating undulations, to the variational absolute minimum.)

Examples

Initial wave functions for the first four states of a one-dimensional particle in a box

• The wave function of the ground state of a particle in a one-dimensional box is a half-period sine wave, which goes to zero at the two edges of the well. The energy of the particle is given by h2n28mL2{displaystyle {frac {h^{2}n^{2}}{8mL^{2}}}}, where h is the Planck constant, m is the mass of the particle, n is the energy state (n = 1 corresponds to the ground-state energy), and L is the width of the well.


• The wave function of the ground state of a hydrogen atom is a spherically symmetric distribution centred on the nucleus, which is largest at the center and reduces exponentially at larger distances. The electron is most likely to be found at a distance from the nucleus equal to the Bohr radius. This function is known as the 1s atomic orbital. For hydrogen (H), an electron in the ground state has energy 3017782103997768000♠−13.6 eV, relative to the ionization threshold. In other words, 13.6 eV is the energy input required for the electron to no longer be bound to the atom.


• The exact definition of one second of time since 1997 has been the duration of 7009919263177000000♠9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom at rest at a temperature of 0 K.^[2]
  

Notes

Bibliography

• 
  Feynman, Richard; Leighton, Robert; Sands, Matthew (1965). "see section 2-5 for energy levels, 19 for the hydrogen atom". The Feynman Lectures on Physics. 3.