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 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.

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

where *V*(*x*) is the potential.

With integration by parts:

Hence in case that is equal to *zero*, one gets:

Now, consider a small interval around ; i.e., . Take a new (deformed) wave function *ψ'*(*x*) to be defined as , for ; and , for ; and constant for . If is small enough, this is always possible to do, so that *ψ'*(*x*) is continuous.

Assuming around , one may write

where is the norm.

Note that the kinetic-energy densities hold everywhere because of the normalization. More significantly, the average kinetic energy is lowered by by the deformation to *ψ'*.

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 *ψ'* because there.

On the other hand, in the interval we have

which holds to order .

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

lower, but still of the same lower order 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 , if we deform the state 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 , 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.)

As the ground state has no nodes it is *spatially* non-degenerate, i.e. there are no two stationary quantum states with the energy eigenvalue of the ground state (let's name it ) and the same spin state and therefore would only differ in their position-space wave functions.^{ [1] }

The reasoning goes by contradiction: For if the ground state would be degenerate then there would be two orthonormal^{ [2] } stationary states and — later on represented by their complex-valued position-space wave functions and — and any superposition with the complex numbers fulfilling the condition would also be a be such a state, i.e. would have the same energy-eigenvalue and the same spin-state.

Now let be some random point (where both wave functions are defined) and set:

- and with (according to the premise
*no nodes*)

Therefore the position-space wave function of is

Hence for all

But i.e. is *a node* of the ground state wave function and that is in contradiction to the premise that this wave function cannot have a node.

Note that the ground state could be degenerate because of different *spin states* like and while having the same position-space wave function: Any superposition of these states would create a mixed spin state but leave the spatial part (as a common factor of both) unaltered.

- 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 , 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 −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 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.
^{ [3] }

- 1 2 See, for example, Cohen, M. (1956). "Appendix A: Proof of non-degeneracy of the ground state" (PDF).
*The energy spectrum of the excitations in liquid helium*(Ph.D.). California Institute of Technology. Published as Feynman, R. P.; Cohen, Michael (1956). "Energy Spectrum of the Excitations in Liquid Helium" (PDF).*Physical Review*.**102**(5): 1189. Bibcode:1956PhRv..102.1189F. doi:10.1103/PhysRev.102.1189. - ↑ i.e.
- ↑ "Unit of time (second)".
*SI Brochure*. International Bureau of Weights and Measures . Retrieved 2013-12-22.

- 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**.

In quantum mechanics, the **Hamiltonian** of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's *energy spectrum* or its set of *energy eigenvalues*, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

In quantum mechanics, the **uncertainty principle** is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, *x*, and momentum, *p*, can be predicted from initial conditions.

The **quantum harmonic oscillator** is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

In physics, an **operator** is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

In quantum mechanics, **perturbation theory** is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system can be expressed as "corrections" to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system.

In condensed matter physics, **Bloch's theorem** states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. Mathematically, they are written:

The **path integral formulation** is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

In physics, a **wave packet** is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant or it may change (dispersion) while propagating.

In physics, the **S-matrix** or **scattering matrix** relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

In mathematical physics, the **WKB approximation** or **WKB method** is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly.

In quantum physics, **Fermi's golden rule** is a formula that describes the transition rate from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time and is proportional to the strength of the coupling between the initial and final states of the system as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.

In physics, the **Lamb shift**, named after Willis Lamb, is a difference in energy between two energy levels ^{2}*S*_{1/2} and ^{2}*P*_{1/2} of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy.

In quantum mechanics, a **two-state system** is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum mechanics, the **momentum operator** is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:

The **Franz–Keldysh effect** is a change in optical absorption by a semiconductor when an electric field is applied. The effect is named after the German physicist Walter Franz and Russian physicist Leonid Keldysh.

In quantum mechanics the **delta potential** is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.

This article relates the Schrödinger equation with the path integral formulation of quantum mechanics using a simple nonrelativistic one-dimensional single-particle Hamiltonian composed of kinetic and potential energy.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

The **Gell-Mann and Low theorem** is a theorem in quantum field theory that allows one to relate the ground state of an interacting system to the ground state of the corresponding non-interacting theory. It was proved in 1951 by Murray Gell-Mann and Francis E. Low. The theorem is useful because, among other things, by relating the ground state of the interacting theory to its non-interacting ground state, it allows one to express Green's functions as expectation values of interaction picture fields in the non-interacting vacuum. While typically applied to the ground state, the Gell-Mann and Low theorem applies to any eigenstate of the Hamiltonian. Its proof relies on the concept of starting with a non-interacting Hamiltonian and adiabatically switching on the interactions.

In quantum mechanics, particularly Perturbation theory, a **transition of state** is a change from an initial quantum state to a final one.

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