To understand the nature of solutions, compare energy to potential at Classically, there are two types of solutions to these equationsBound States are when E < V().Unbound states are when E > V()Quantum mechanically, there are bound and unbound states as well, with the same criteriaBound states are a little easier to understand, so well do these firstE > V()E < V()V(x)Bound and Unbound States

Wave Function ConstraintsThe wave function (x) must be continuousIts derivative must exist everywhereIts derivative must be continuous*Its second derivative must exist everywhere*The wave function must be properly normalizedMust not blow up at *Except where V(x) is infiniteThere are exceptions and ways around this problemNormalizationWhat if the wave function is not properly normalized?If the integral is finite, multiply by a constant to fix itModified wave satisfies SchrdingerIf the integral is infinite, it gets complicatedFor bound states, this is still troubleFor unbound states, it is okay

The 1D infinite square well (1)Outside of the well, the wave function must vanishIn remaining region, we need to solve differential equationWhat functions are minus their second derivative?We prefer realBoundary ConditionsMust vanish at x = 0And at x = LWhere does sin vanish?Dont worry about derivative because V(x) blows up thereV(x)

The 1D infinite square well (2)mL2E/2Energy Diagramn = 1n = 2n = 3n = 4

1D Infinite Square Well (3)BUT WAIT: What about normalization?To fix it: multiply by (2/L).The most general solution is superposition of this solution

The Finite Square Well (1)We need to solve equation in all three regions; this takes workIn region II, we get solutions like beforeNo longer necessary that it vanish at the boundariesIn regions I and III, we solve a different equation:If we are looking at bound state (E < V0), in these regions, we get exponentialsIIIIII

The Finite Square Well (2)Wave function must not blow up at Wave function must be continuous at x = LIIIIIIDerivative of wave function must be continuous at x = L

The Finite Square Well (2)Wave function must not blow up at Wave function must be continuous at x = LIIIIIIDerivative of wave function must be continuous at x = L

The Finite Square Well (3)Wave function penetrates into forbidden regionOscillates when E > V(x)Damps when E < V(x)Energies decreased slightly compared to infinite square wellFinite number of bound statesDue to finite extension and depth of potential wellEnergy DiagramInfinite Welln = 4n = 3n = 2n = 1Solve all these equations simultaneouslyNormalize the final wave function

The Harmonic Oscillator (1)At large x, the behavior is governed (mostly) by the x2 termNow we guessDont want it blowing up at infinity!Our strategy:Check that this worksFind more solutions and check them

The Harmonic Oscillator (2)

n = 3n = 2n = 1n = 0The Harmonic Oscillator (3)We still need to normalize itExpect other solutions to have similar behavior, at least at large xWe will guess the nature of these solutionsMultiply the wave function above by an arbitrary polynomial P(x)Substitute in and see if it worksThis will give you En