The timedependent wave function the evolution of the ground state of the harmonic oscillator in the presence of a timedependent driving force has an exact solution. In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve. Schrodingers equation and the ground state wave function. First of all, the ground state wave function of the harmonic oscillator represents the minimum uncertainty state, for which the heisenberg uncertainty inequality for momentum and position x p. The simple harmonic oscillator, a nonrelativistic particle in a potential 2 1. An harmonic oscillator is a particle subject to a restoring force that is proportional to the displacement of the particle. Since the ho potential has a parity symmetry, the solutions either have even or odd parity. These sets are less common in mathematical physics than the legendre and bessel functions of chapters 11 and 12, but hermite polynomials occur in solutions of the simple harmonic oscillator of quantum mechanics and laguerre polynomials in wave functions of the hydrogen atom.
The 1d harmonic oscillator the harmonic oscillator is an extremely important physics problem. As a gaussian curve, the ground state of a quantum oscillator is. The stationary states of the harmonic oscillator have been considered already in chapter 2 where the corresponding wave. This is a gaussian minimum uncertainty distribution. The wavefunctions for the quantum harmonic oscillator contain the gaussian form which allows them to satisfy the necessary boundary conditions at infinity. Using the number operator, the wave function of a ground state harmonic oscillator can be found. What we have shown so far is that h minjhj mini h 0 jhj 0ie 0 29 where j 0iis the ground state energy. I then introduce hermite polynomials which are used to find the first and second excited states. The simple harmonic oscillator michael fowler 116 einsteins solution of the specific heat puzzle the simple harmonic oscillator, a nonrelativistic particle in a potential 2 1 2 kx, is an excellent model for a wide range of systems in nature. For the harmonic oscillator potential in the timeindependent schrodinger equation. Harmonic oscillator solution using operators operator methods are very useful both for solving the harmonic oscillator problem and for any type of computation for the ho potential. To the extent that the oscillators wave function extends beyond the classical turning point, tunneling is occurring.
The harmonic oscilllator wavefunctions are the eigenfunctions of a one dimensional hamiltonian operator with a quadratic potential energy. Repetitively applying the raising operator to the ground state wave function then allows the derivation of the general formula describing wave. With this convention, the normalized wave functions, namely, the stationary states of. Mar 04, 2007 homework statement use the ground state wave function of the simple harmonic oscillator to find. Note that although the integrand contains a complex exponential, the result is real. But its still a 3d state that has a lower energy than the state thats usually called the ground state of the 3d sho. One of the most important problems in quantum mechanics is the simple harmonic oscillator, in part. The linear harmonic oscillator is described by the schrodinger equation.
It is clear that the center of the wave packet follows the motion of a classical 2d isotropic harmonic oscillator, i. The simple harmonic oscillator, a nonrelativistic particle in a potential. The calculation below shows that the probability that tunneling occurs is independent of the values of k and \\mu\ for the ground state. 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. The quantum harmonic oscillator part 2 finding the wave. Harmonic oscillator zeropoint energy from uncertainty principle4 this is the lowest possible value for the energy, but is it actually the ground state energy. Harmonic oscillator a we rewrite the hamiltonian h p 2. The half harmonic oscillators ground state wave function. The normalised ground state positionspace wave function of the harmonic oscillator has the form a determine and a in terms of m, w, h. The simple harmonic oscillator asaf peer1 november 4, 2015 this part of the course is based on refs. An example of such a parametric form for a symmetric well ground state centered about the origin might be a gaussian distribution simple harmonic oscillator ground state of the form. You are correct in that for any given harmonic oscillator we can define the zero of the energy so that the ground state has zero energy. H p2 2m 1 2 m 2 x2 this hamiltonian appears in various applications, and in fact the approximation of the harmonic oscillator is valid near the minimum of any potential function. That is, x0 is the classical turning point of the oscillation when the oscillator wavefunction has 1 loop.
The harmonic oscillator is characterized by the hamiltonian. Plus, that state has l0 too as i mentioned, it corresponds to the first excited state of the equivalent 1d oscillator. Check the result by using the similarity between the schrodinger equations in position and. Expanded around a minimum point x, any potential can then be taylor expanded as. Coming from a classical perspective, its still a curiosity. This wave function can be split into a spatial and spin part. Note that since, the normalization condition is despite this, because the potential energy rises very steeply, the wave functions decay very rapidly as increases from 0 unless is very large. The quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. Expanding an arbitrary potential energy function in a taylor series, where x 0 is the. Harmonic oscillator wave functions and probability density. Classical limit of the quantum oscillator a particle in a quantum harmonic oscillator in the ground state has a gaussian wave function. Harmonic oscillator i lecture 8 physics 342 quantum mechanics i wednesday, february 10th, 2010. Almost all potentials in nature have small oscillations. Consider a macroscopic simple harmonic oscillator, and to keep things simple assume there are.
The ground state of a quantum mechanical harmonic oscillator. The operators we develop will also be useful in quantizing the electromagnetic field. Harmonic motion is one of the most important examples of motion in all of physics. The normalized wave functions in terms of dimensional less parameter. At a couple of places i refefer to this book, and i also use the same notation, notably xand pare operators, while the correspondig eigenkets are jx0ietc. Therefore, the correctly normalized ground state wave function is. Recall that the maximum or minimum of a function fx occurs when dfdx 0. In classical physics this means f mam 2 x aaaaaaaaaaaaa t2 kx. It is useful to exhibit the solution as an aid in constructing approximations for more complicated systems. Furthermore, it is one of the few quantummechanical systems. Feb 04, 2017 in the second video out of 3 in this series, we solve for the exact wave function for the ground state.
The potential for the harmonic ocillator is the natural solution every potential with small oscillations at the minimum. Harmonic oscillator physics lecture 9 physics 342 quantum mechanics i friday, february 12th, 2010. Any vibration with a restoring force equal to hookes law is generally caused by a simple harmonic oscillator. Therefore, the correctly normalized ground state wave function is y0 hxl h. Nonzero ground state energy of the quantum harmonic oscillator. Introduction we return now to the study of a 1d stationary problem. Harmonic oscillator physics lecture 9 physics 342 quantum mechanics i friday, february 12th, 2010 for the harmonic oscillator potential in the timeindependent schr odinger equation. Short lecture on the harmonic oscillator wavefunctions. Substituting this function into the schrodinger equation by evaluating the second derivative gives. A general wavefunction of the sho is a superposition or linear combination of its eigenfunctions. Similarly, the wave functions will be products of onedimensional harmonic oscillator functions in the, and directions. At what displacement is the oscillator most likely to be found in its ground state.
Most of the time the particle is in the position x0 since there the velocity is zero, while at x 0 the velocity is maximum. One of the most important problems in quantum mechanics is the simple harmonic oscillator, in part becauseitspropertiesaredirectlyapplicableto. Ground state wavefunction of two particles in a harmonic. Ground state solution to find the ground state solution of the schrodinger equation for the quantum harmonic oscillator. Consider a macroscopic simple harmonic oscillator, and to keep things simple assume there are no interactions with the rest of the universe. As a gaussian curve, the ground state of a quantum oscillator is how. This is of both an extreme importance in physics, and is very. The possible energy states of the harmonic oscillator potential v form a ladder of even and odd wave functions with energy di erences of the ground state is a gaussian distribution with width x. Find the momentumspace wave function by fourier transformation. Thus, the ground state would be thus, the ground state would be and other wave functions can be constructed in a similar manner. Normalizing the quantum harmonic oscillator wave function. Schrodingers equation 2 the simple harmonic oscillator.
What is the effect of the lowering operator on the ground state, eg. The harmonic oscillator energy levels are equallyspaced, by. The wave function of a quantum harmonic oscillator varies depending on the energy level of the particle being described. The hookes atom is a simple model of the helium atom using the quantum harmonic oscillator. This means that when 1 h 35cl is in its ground state its classically allowed region is 2 x0 0. To solve the harmonic oscillator equation, we will first change to dimensionless variables, then find the form of the solution for, then multiply that solution by a polynomial, derive a recursion relation between the coefficients of the polynomial, show that the polynomial series must terminate if the solutions are to be normalizable, derive the energy eigenvalues, then finally derive the functions that are solutions. We will study in depth a particular system described by the h. Being an antisymmetric wave function, when the spatial part is symmetric the spin part is antisymmetric and vice versa.
Thus, the ground state would be and other wave functions can be constructed in a similar manner. In a given state, the normalized wave function of the electron is given byx 2l sin 2. In the wavefunction associated with a given value of the quantum number n, the gaussian is multiplied by a polynomial of order n. In the wavefunction associated with a given value of the quantum number n, the gaussian is multiplied by a polynomial of order n the hermite polynomials above and the constants necessary. This phenomenon is called the zeropoint energy or the zeropoint motion, and it stands in direct contrast to the classical picture of a vibrating molecule. How to find the wave function of the ground state of a. Wave functions, allowed energies, and corresponding probability densities for the harmonic oscillator.
The potential energy, v x in a 1d simple harmonic oscillator. For the half harmonic oscillator the ground state wave function is of the form xexpx22 but sir how to check its parity and with respect to with point as this function is valid for positive x only thank you. There are several items of note that should be verified by the reader. Many potentials look like a harmonic oscillator near their minimum.
428 464 265 48 317 569 775 761 777 1038 397 218 1001 427 539 3 508 377 1026 1309 440 27 844 930 603 419 540 118 910 161 1094 852 37 1410 1305