149Apply separation of variables to the 149Consider general characteristics of the 149Solve the infinite square well x000E x Uxxti Solving the TDSE We assume that the solut ID: 833201
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Physics 48 –January 30, 2008•
Physics 48 –January 30, 2008•Apply separation of variables to the •Consider general characteristics of the •Solve the infinite
square well. ()()â=xUxxt
square well. ()()â=xUxxti,,,Solving the TDSEWe assume that the solution is separable. i.e. xtxft,() ()()tfxitfxxUxtf)()()(
)(Divide by xtxft,()This yields tftf
)(Divide by xtxft,()This yields tftfxUxm)()(22=Left side only depends on , right only depends on Separating the equationsTo be true fo
r all times and positions we require tha
r all times and positions we require that both sides equal a constant. Call it tftfxUxm)()(22=So )()(tftdf(Ordinary differential equati
on!) Which has a solution )(tiEetfIf we
on!) Which has a solution )(tiEetfIf we identify then fte. And for the time dependent part tftfxUxm)()(22=must satisfy the TISE ()()(
)xExxUxd\\=22Solving the Time-i
)xExxUxd\\=22Solving the Time-independent Schrodinger Write the SE with the appropriate potential in all regions of space. Solve the resulti
ng differential equation in each region
ng differential equation in each region of space. Apply boundary conditions (this usually determines the allowed energies). For discontinuous potenti
als – require continuity of the wav
als – require continuity of the wave function, and usually its derivative. Evaluate all undetermined constants. Normalize – the probability
of finding the particle somewhere shoul
of finding the particle somewhere should be one. Discard any solutions with infinite probabilities (they are non-physical). Calculate the expectatio
n values of the quantities of interest.
n values of the quantities of interest. dxxxfxfââ<=)()(Particle in a BoxWith ) = 0 if and Ux if x0 or if x� L Inside the box we
have 2222mxxEx or ()()Exkx=â{â
have 2222mxxEx or ()()Exkx=â{âThe equation is that of a harmonic oscillator with solutions xAkxBkxsincosWe match these solutions with
in the well to the Boundary Condition th
in the well to the Boundary Condition that the wavefunction must vanish a 0 and at L. These imply that B=0 and that the constant knL, with n an in