This encourages us to try to find a total solution ψ( y) by a function of the formĭivide through by −ψ 0( y)/2 and we find the equation for f We conclude that ψ 0( y) ≡ exp(- y 2/2) is an eigenfunction Note that all constants are hidden in the equation on the right.Ī particular solution of this equation is Secondly, we divide through by, a factor that has dimension energy, As a first step we define the angular frequency ω ≡ √ k/ m, the same formula as for the classical harmonic oscillator: We rewrite the Schrödinger equation in order to hide the physical constants m, k, and h. Note that the n = 2 wave function (black) is not zero in these regions. The same holds for the region to the right of the small black line at positive x. As an example two small vertical lines are plotted: to the left of the small line that lies at negative x the potential energy is larger than E 2, the energy of the n = 2 level. In the figure we see that the probability of finding the oscillating mass at points to the left (for negative x) or to the right (for positive x) of V( x) is non-zero and we notice that for these points the potential energy V( x) is larger than the energy eigenvalue. As an example of a zero level, the zero (asymptotic) energy of the n = 2 function (black) is plotted as the black horizontal line of constant energyĪccording to quantum mechanics, the wave function squared of a point mass, |ψ( x)|², is the probability of finding the mass in the point x. The functions are shifted upward such that their energy eigenvalues coincide with the asymptotic levels, the zero levels of the wave functions at x = ±∞. Note that the lowest function (blue) has indeed the form of a Gaussian function. The four lowest energy harmonic oscillator eigenfunctions are shown in the figure. Quantum mechanically this is not possible, the position x of the particle of lowest possible energy is described by a wave function (a Gaussian function) of energy ½hν. Classically, the oscillating particle can come to rest at the bottom of the potential well, where it has zero kinetic and potential energy and a well-defined-sharp-position. Hence, by Heisenberg's uncertainty relation, energy and position cannot be sharp simultaneously.
This well-defined, non-vanishing, zero-point energy is due to the fact that the position x of the oscillating particle cannot be sharp (have a single value), since the operator x does not commute with the energy operator. Note that for n = 0 the energy E 0 is not equal to zero, but equal to the zero-point energy ½ hν.
Where h is Planck's constant and ν depends on the the mass and the stiffness of the oscillator (see below). The corresponding energy eigenvalues are labeled by a single quantum number n, As stated above, the Schrödinger equation of the one-dimensional quantum harmonic oscillator can be solved exactly, yielding analytic forms of the wave functions (eigenfunctions of the energy operator). The characterizing feature of the one-dimensional harmonic oscillator is a parabolic potential field that has a single minimum usually referred to as the "bottom of the potential well". Also the energy of electromagnetic waves in a cavity can be looked upon as the energy of a large set of harmonic oscillators. Two well-known examples are the vibrations of the atoms in a diatomic molecule about their equilibrium position and the oscillations of atoms or ions of a crystalline lattice. Whenever one studies the behavior of a physical system in the neighborhood of a stable equilibrium position, one arrives at equations which, in the limit of small oscillations, are those of a harmonic oscillator. In quantum mechanics, the one-dimensional harmonic oscillator is one of the few systems that can be treated exactly, i.e., its Schrödinger equation can be solved analytically.Īlthough the harmonic oscillator per se is not very important, a large number of systems are governed approximately by the harmonic oscillator equation. The prototype of a one-dimensional harmonic oscillator is a mass m vibrating back and forth on a line around an equilibrium position.