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notes03_qm3_particle_in_a_box

# notes03_qm3_particle_in_a_box - 3 Bound States The...

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3 Bound States The Schr¨ odinger equation is a second order differential equation because it contains the second derivative of the wavefunction. For a second order equation, the solutions have two degrees of freedom. In the case of a free particle, there were no boundary conditions, so that the wavefunction had both degrees of freedom. One is the energy (or wavelength), which could take any positive value. The second is the phase of the wave. If a single wall is introduced (for example at x = 0), the value of the wavefunction is constrained at the wall, and the phase is restricted, leaving only one degree of freedom - the energy. We can see this by looking at the general solution (in terms of sin and cos) ψ ( x ) = A sin( kx ) + B cos( kx ) (3.1) and imposing the constraint that ψ (0) = 0 at the wall (at x = 0). To satisfy this condition, there can be no cos component, so that B = 0 and the solution for x > 0 is ψ ( x ) = A sin( kx ) . (3.2) In this section, we will look at the simplest example of a potential which imposes two constraints on the wavefunction. With two constraints, the second order Schr¨ odinger equation fully defines the allowed wavefunctions. With no degrees of freedom, the wavefunction and its energy are restricted to precise, quantized values. 3.1 Particle in a box A potential energy defined by, V ( x ) = 0 for 0 x L for x < 0 and x > L (3.3) is a box that constrains a particle between x = 0 and x = L . Within this box, the potential is constant, and the particle is free. At the boundaries of the box the potential becomes infinite, so that the wavefunction must become zero. This can be seen by looking at the Schr¨ odinger equation 2 2 m ∂ψ ( x ) ∂x + V ( x ) ψ ( x ) = ( x ) . (3.4) To have a finite energy value, the second term, V ( x ) ψ ( x ), must be well behaved. If

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