Ground state

From formulasearchengine
Jump to navigation Jump to search

{{#invoke:Hatnote|hatnote}}

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. The ground state of a quantum field theory 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 which 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.

1D ground state has no nodes

In 1D the ground state of the Schrödinger equation has no nodes. This can be proved considering an average energy in the state with a node at , i.e. . Consider the average energy in this state

where is the potential. Now consider a small interval around , i.e. . Take a new wavefunction to be defined as and and constant for . If epsilon is small enough then this is always possible to do so that is continuous. So assuming around , we can write the new function as

where is the norm. Note that the kinetic energy density everywhere because of the normalization. Now consider the potential energy. For definiteness let us choose . Then it is clear that outside the interval the potential energy density is smaller for the because there. On the other hand in the interval we have

which is correct to this order of and indicate higher order corrections. On the other hand the potential energy in the state is

which is the same as that of the state to the order shown. Therefore the potential energy unchanged to leading order in by deforming the state with a node into a state without a node . We can do this by removing all nodes thereby reducing the energy, which implies that the ground state energy must not have a node. This completes the proof.

Examples

Initial wave functions for the first four states of a one-dimensional particle in a box
  • The exact definition of one second of time since 1997 has been the duration of 9,192,631,770 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.[1]

Notes

Bibliography

  • {{#invoke:citation/CS1|citation

|CitationClass=book }}