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The zero-point energy makes no contribution to Planck's original law, as its existence was unknown to Planck in 1900. [25] The concept of zero-point energy was developed by Max Planck in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900. [26]
Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced by ħω, the Morse potential level spacing decreases as the energy approaches the dissociation energy. The dissociation energy D e is larger than the true energy required for dissociation D 0 due to the zero point energy of the lowest (v = 0) vibrational level.
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 stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system.
The energy levels increase with , meaning that high energy levels are separated from each other by a greater amount than low energy levels are. The lowest possible energy for the particle (its zero-point energy) is found in state 1, which is given by [10] = =.
An equivalent formula can be derived quantum mechanically from the time-independent Schrödinger equation with a kinetic energy Hamiltonian operator using a wave function as an eigenfunction to obtain the energy levels as eigenvalues, but the Rydberg constant would be replaced by other fundamental physics constants.
It is convenient to define a characteristic vibrational temperature , = where is experimentally determined for each vibrational mode by taking a spectrum or by calculation. By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes = =, /
+ = + = Multiplying the first equation by and the second by and subtracting one from the other, we get: = Integrating both sides = In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: = where is, in general, a complex constant.
The Planck energy E P is approximately equal to the energy released in the combustion of the fuel in an automobile fuel tank (57.2 L at 34.2 MJ/L of chemical energy). The ultra-high-energy cosmic ray observed in 1991 had a measured energy of about 50 J, equivalent to about 2.5 × 10 −8 E P .