Harmonic oscillators zero-point energy

Tlris is the Schrodinger equation for a simple harmonic oscillator. The energies of the system are given by E = (i + ) x liw and the zero-point energy is Hlj. 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

The vibrational levels corresponding to n = 0,1,2... are evenly spaced. Like the particle confined to a line segment, the harmonic oscillator also has zero-point energy Eq = hu. 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

Fig. 4.1 The zero point energy or low temperature approximation As temperature drops and u increases above u 4 the harmonic oscillator partition function Q (Harm. Osc.) is better and better approximated by the zero point energy term, exp(—u/2). For a typical CH stretching frequency, v = 3000 cm-1, u 4 at 1050 K and it is reasonable to use the ZPE approximation for that frequency at temperatures below 1000 k...

Zero-point energies are obtained from vibrational spectra using experimental frequencies whenever available, while the inactive frequencies are extracted from data calculated by means of an appropriate force-held model. In the harmonic oscillator approximation, the zero-point energy is... 

The zero point energy of a simple harmonic oscillator is ... 

A harmonic oscillator (1 dimension) has a zero point energy ... 

Within the harmonic oscillator approximation, the energy of the lowest vibrational level can be determined from Eq. (9.47) as ha>/2 where h is Planck s constant (6.6261 x 10- J s) and a> is the vibrational frequency. The sum of all of these energies over all molecular vibrations defines the zero-point vibrational energy (ZPVE). We may then define the internal energy at 0 K for a molecule as... 

It is also interesting to note that the lowest allowed energy level, with n -1 in eqn 2.32, is not zero. Zero-point energy arises in some other situations, such as the harmonic oscillator. As explained in Section 3.2, it is related to the uncertainty principle, discussed below. 

One interesting prediction of eqn 3.21 is that the lowest possible energy for a harmonic oscillator is nor zero. The value of E0 given by eqn 3.18 is called the zero-point energy. A similar result was found for the particle in a box (see eqn 2.32 in Section 2.3). 

Even if there is no electromagnetic field present, the vector potential exhibits fluctuations A = (4 ) + 84, so that even if there is only the vacuum, physics still involves this fluctuation. This is also seen in the zero-point energy of the harmonic oscillator expansion of the fields. So an electron will interact with virtual photons. If we represent all of these interactions as a blob coupled to the path of an electron, this blob may be expanded into a sum of diagrams where the electron interacts with photons. Each term is an order expansion and contributes... 

The term including zero-point energy (o0l2. In the classically accessible region near the minima x0 one may use the harmonic oscillator approximation. The factor N is determined in a manner similar to the route described previously for a metastable potential. The tunneling splitting calculated from (A.23) is... 

In a simple approach that treats chemical bonds as harmonic oscillators, the vibrational energy of a molecule at absolute zero, the so-called zero-point energy, is given by... 

Fig. 1.1.2 The energy levels of a one-dimensional harmonic oscillator. The zero-point energy Eij = huj/2, where for a diatomic molecule oj = Jk///, k is the force constant, and /./ is the reduced mass.

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