qmparticles_in_potentials_03222011

qmparticles_in_potentials_03222011 - Quantum Mechanics...

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Quantum Mechanics: Particles in Potentials 3 april 2010 I. Applications of the Postulates of Quantum Mechanics Now that some of the machinery of quantum mechanics has been assem- bled, one can begin to apply the concepts of wavefunctions, superposition, eigenfunctions, operators, eigenvalues, observables, etc. to derive wavefunc- tions for several model systems. These will aFord insight in to some fun- damental ideas to be extended later in the context of more complicated systems. The ±rst system will be a free particle, i.e., a particle with no exter- nal potential acting on it. The second system will be a particle in a one- dimensional box with a de±ned external potential, V ( x ). 1. ²ree Particle Consider a free particle, i.e., one that does not have an external poten- tial acting upon it. In the development of the time-independent Schrodinger equation discussed earlier, the total energy of a quantum mechanical state is taken to be composed of a Kinetic and a Potential component. In this ±rst example, consider the potential is constant everywhere; in this case, there is no spatial dependence of the potential. Moreover, since the potential is everywhere equal, we can allow it to be null everywhere. Thus, the time-independent Schrodinger equation becomes, - ~ 2 2 m 2 ψ ( x ) ∂x 2 = ( x ) Rearranging, we have: 2 ψ ( x ) ∂x 2 = - 2 m ~ 2 ( x ) 1
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Solutions to this equation are of the form: ψ + ( x ) = A + e + i 2 mE/ ~ 2 x = A + e + ikx ψ - ( x ) = A - e - i 2 mE/ ~ 2 x = A - e - ikx Here the relation k = 2 π/λ = p 2 mE/ ~ 2 has been used; this is obtained via: k = 2 π λ = 2 π p h E = p 2 2 m p = 2 mE This gives us for k : k = 2 π h 2 mE k = 1 ~ 2 mE k = r 2 mE ~ 2 To obtain the time-dependent wavefunction, one can multiply the spatially- dependent function by the time dependent e - iωt . Note: Solutions are plane waves (moving in + x and - x directions) Note: This result is analogous to the classical solution to a free particle moving in zero external Feld with constant velocity . This results from 2
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the fact that the wavevector, k = 2 π/λ = p 2 mE/ ~ 2 = mv/ ~ is constant. Thus, the velocity of the wavefunction has to be constant and equal to the initial speciFed velocity. Note: The Energy can take on all values ( k is continuous). Thus, the quantum mechanical free particle can take on a continuous spectrum of en- ergies. The free particle energies are not quantized Keep in mind that for the free particle, we have NOT imposed any BOUNDARY con- ditions or restrictions on the behavior of the wavefunction, apart from the requirement that it satisfy a second-order diFerential equation .
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This note was uploaded on 05/10/2011 for the course CHEM 444 taught by Professor Dybowski,c during the Spring '08 term at University of Delaware.

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qmparticles_in_potentials_03222011 - Quantum Mechanics...

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