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## Back To Solutions of Schrödinger's equa.

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**Particle in a Box**E4 Energy E3=9E1 E2=4E1 E1 E=0**Particle in a Box (Quantization of Momentum)**• Using the momentum operator we can determine the avg momentum.**Particle in a Box (Quantization of Momentum)**• Using the momentum operator we can determine the avg momentum.**Particle in a Box (Quantization of Momentum)**• Using the momentum operator we can determine the avg momentum.**Particle in a Box (Quantization of Momentum)**• Integrating by parts we get**Particle in a Box (Quantization of Momentum)**• Integrating by parts we get**Particle in a Box (Quantization of Momentum)**• Integrating by parts we get • Hence avg. momentum =0**Particle in a Box (Finite Square Well)**• The infinite potential is an oversimplification which can never be realised.**Particle in a Box (Finite Square Well)**• The infinite potential is an oversimplification which can never be realised. • A more realistic finite square well is shown U E 0 L**Particle in a Box (Finite Square Well)**• The infinite potential is an oversimplification which can never be realised. • A more realistic finite square well is shown Regions of interest are shown (1-3) U E 1 2 3 0 L**Particle in a Box (Finite Square Well)**• Practical, given enough energy a particle can escape any well.**Particle in a Box (Finite Square Well)**• Practically, given enough energy a particle can escape any well. • A classical particle with E>U can enter the well where it moves freely with a reduced energy (E-U).**Particle in a Box (Finite Square Well)**• Practically, given enough energy a particle can escape any well. • A classical particle with E>U can enter the well where it moves freely with a reduced energy (E-U). • A classical particle with E<U is trapped within the well (can not escape).**Particle in a Box (Finite Square Well)**• Practically, given enough energy a particle can escape any well. • A classical particle with E>U can enter the well where it moves freely with a reduced energy (E-U). • A classical particle with E<U is trapped within the well (can not escape).**Particle in a Box (Finite Square Well)**• In QM there is a probability of it being outside the well.**Particle in a Box (Finite Square Well)**• In QM there is a probability of it being outside the well. That is the waveform is nonzero outside the region 0<x<L.**Particle in a Box (Finite Square Well)**• In QM there is a probability of it being outside the well. That is the waveform is nonzero outside the region 0<x<L. • For stationary states is found from the time independent Schrödinger’s equation.**Particle in a Box (Finite Square Well)**• In QM there is a probability of it being outside the well. That is the waveform is nonzero outside the region 0<x<L. • For stationary states is found from the time independent Schrödinger’s equation.**Particle in a Box (Finite Square Well)**• In QM there is a probability of it being outside the well. That is the waveform is nonzero outside the region 0<x<L. • For stationary states is found from the time independent Schrödinger’s equation.**Particle in a Box (Finite Square Well)**• Therefore outside the well where U(x)=U,**Particle in a Box (Finite Square Well)**• Therefore outside the well where U(x)=U, • where**Particle in a Box (Finite Square Well)**• Therefore outside the well where U(x)=U, • where • Since U>E, this term is positive.**Particle in a Box (Finite Square Well)**• Therefore outside the well where U(x)=U, • where • Since U>E, this term is positive. • Independent solns to the differential are**Particle in a Box (Finite Square Well)**• To keep the waveform finite as and**Particle in a Box (Finite Square Well)**• To keep the waveform finite as and • Therefore the exterior wave takes the form**Particle in a Box (Finite Square Well)**• To keep the waveform finite as and • Therefore the exterior wave takes the form • The internal wave is given as before by**Particle in a Box (Finite Square Well)**• The coefficients are determined by matching the exterior wave smoothly onto the wavefunction for the well interior.**Particle in a Box (Finite Square Well)**• The coefficients are determined by matching the exterior wave smoothly onto the wavefunction for the well interior. • That is, and are continuous at the boundaries.**Particle in a Box (Finite Square Well)**• The coefficients are determined by matching the exterior wave smoothly onto the wavefunction for the well interior. • That is, and are continuous at the boundaries. • This is obtained for certain values of E.**Particle in a Box (Finite Square Well)**• Because is nonzero at the boundaries, the de Broglie wavelength is increase and hence lowers the energy and momentum of the particle.**Particle in a Box (Finite Square Well)**• The solution for the finite well is**Particle in a Box (Finite Square Well)**• The solution for the finite well is • As long as is small compared to L.**Particle in a Box (Finite Square Well)**• The solution for the finite well is • As long as is small compared to L. • where**Particle in a Box (Finite Square Well)**• The solution for the finite well is • As long as is small compared to L. • where • The approximation only works for bounds states. And is best for the lowest lying states.**The Quantum Oscillator**• We now look at the final potential well for which exact results can be obtained.**The Quantum Oscillator**• We now look at the final potential well for which exact results can be obtained. • We consider a particle acted on by a linear restoring force .**The Quantum Oscillator**• We now look at the final potential well for which exact results can be obtained. • We consider a particle acted on by a linear restoring force . -A A X=0**The Quantum Oscillator**• We now look at the final potential well for which exact results can be obtained. • We consider a particle acted on by a linear restoring force . • x rep. its displacement from stable equilibrium(x=0).**The Quantum Oscillator**• We now look at the final potential well for which exact results can be obtained. • We consider a particle acted on by a linear restoring force . • x rep. its displacement from stable equilibrium(x=0). • Strictly applies to any object limited to small excursions about equilibrium.**The Quantum Oscillator**• The motion of a classical oscillator with mass m is SHM at frequency**The Quantum Oscillator**• The motion of a classical oscillator with mass m is SHM at frequency • If the particle is displaced so that it oscillated between x=A and x=-A, with total energy the particle can be given any (nonnegative) energy including zero.**The Quantum Oscillator**• The quantum oscillator is described by introducing the potential energy into Schrödinger's equation.**The Quantum Oscillator**• The quantum oscillator is described by introducing the potential energy into Schrödinger's equation. • So that,**The Quantum Oscillator**• The quantum oscillator is described by introducing the potential energy into Schrödinger's equation. • So that,**The Quantum Oscillator**• The quantum oscillator is described by introducing the potential energy into Schrödinger's equation. • So that,**The Quantum Oscillator**• We can consider properties which our wavefunction must and can’t have.