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#Quantum harmonic oscillator series#
Series of y as the general solution to equation (13). So that we can find an exact solution for ( y). The next step is to solve the second order differential equation (13) above for u(y) Putting the values from (8) and (10) in equation (7): We must now calculate the derivatives of that will be substitued into the With this fact we can guess that ( y) will be as e ±. Substituting this new variable into Equation (5) above yeilds:įor very large values of y, the term is negligible in comparison to the y 2 term. The first step in the power series method is to perform aĬhange of variables by introducing the dimensionless variable, Series method is used to derive the wave function and the eigenenergies for the The Equation for the Quantum Harmonic Oscillator is a second order differentialĮquation that can be solved using a power series. Placing this potential in the one dimensional, time-independent Schr ödinger Where is the natural frequency, k is the spring constant, and m is the mass of theįor convenience in this calculation, the potential for the harmonic oscillator is The classical potential for a harmonic oscillator is derivable from Hooke’s law. A firm understanding of the principles governing the harmonic oscillator is prerequisite to any substantial study of quantum mechanics.ġ.1 The Schrodinger Equation for the Harmonic Oscillatorģ Solved Harmonic Oscillator Problems 1 Solution of the Schrodinger Equation The solution to this simple system can then be used on them. Systems with nearly unsolvable equations are often broken down into smaller systems. It is one of the first applications of quantum mechanics taught at an introductory quantum level. An exact solution to the harmonic oscillator problem is not only possible, but also relatively easy to compute given the proper tools. The equation for these states is derived in section 1.2. The harmonic oscillator has only discrete energy states as is true of the one-dimensional particle in a box problem. This equation is presented in section 1.1 of this manual. The Harmonic Oscillator is characterized by the its Schr ö dinger Equation.
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Here, harmonic motion plays a fundamental role as a stepping stone in more rigorous applications. Almost all potentials in nature have small oscillations at the minimum, including many systems studied in quantum mechanics. The potential for the harmonic ocillator is the natural solution every potential with small oscillations at the minimum. Any vibration with a restoring force equal to Hooke’s law is generally caused by a simple harmonic oscillator. The leading non-constant term is in the form of a harmonic oscillator, and thus this potential can be approximately treated as a harmonic oscillator.Harmonic motion is one of the most important examples of motion in all of physics. If the original potential energy is symmetric about x = 0, we can expand about x 0 = 0 to yield Problem 12.7: Determining the properties of half wells.ġA generic potential energy function, V( x), can be expanded in a Taylor series to yield.Problem 12.6: Determining the properties of half wells.Problem 12.5: Describe the effect of the added potential energy function.Problem 12.4: A particle is confined to a box with an added unknown potential energy function.Problem 12.3: Two-state superpositions in the harmonic oscillator.Problem 12.2: A particle is in a 1-d dimensionless harmonic oscillator potential.Problem 12.1: Compare classical and quantum harmonic oscillator probability distributions.Section 12.6: Exploring Other Spatially-varying Wells.Section 12.5: Ramped Infinite and Finite Wells.Section 12.3: Classical and Quantum-Mechanical Probabilities.Section 12.2: The Quantum-mechanical Harmonic Oscillator.Section 12.1: The Classical Harmonic Oscillator.Several systems in nature exactly exhibit the harmonic oscillator's potential energy, but many more systems approximately exhibit the form of the harmonic oscillator's potential energy. We begin with the most recognizable of these problems, that of the simple harmonic oscillator, V( x) = mω 2 x 2/2, is perhaps the most ubiquitous potential energy function in physics. In this chapter we will consider eigenstates of potential energy functions that are spatially varying, V( x) ≠ constant. Chapter 12: Harmonic Oscillators and Other Spatially-varying Wells