3. A ID quantum well is described by the following position dependent potential energy for a moving particle of mass m: V = {-V, 0sx L, where V, is a positive constant and L is the width of the quantum well. The total energy E of the particle is between -V, and 0 (i.e., -V, < E < 0 so bound states only!). Find the wavefunctions in each region. Apply as many boundary conditions as possible to these wavefunctions. Normalization of the wavefunctions is not required.

Suppose that you have a 2D quantum system where X and Px are the x- component position and momentum operators and Y and Py are the y- component position and momentum operators. Which of the following commutators is not equal to 0? [Py,Y] O IX,Y] O [Px,Px] O [PxY]

The average lifetime of a Z boson is about 3.0 x 10 J Need Help? Read It -25 s. Estimate the minimum uncertainty in the energy of a Z boson.

Q3: For a quantum harmonic oscillator in its ground state. Find: a) (x) b) (x²) с) Ох

3. Plane waves and wave packets. In class, we solved the Schrodinger equation for a "free particle" (e.g. when U(x,t) = 0). The correct[solution is (x, t) = Ae(px-Et)/ħ This represents a "plane wave" that exists for all x. However, there is a strange problem with this: if you try to normalize the wave function (determine A by integrating * for all x), you will find an inconsistency (A has to be set equal to 0?). This is because the plane wave stretches to infinity. In order to actually represent a free particle, this solution needs to be handled carefully. Explain in words (and/or diagrams) how we can construct a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane waves for a limited range of momentum/energy. Hint 2: have a look at the brief discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)

Subject Quantum Mechanics.  Wave function normalization and superposition of solutions. Wavel functions, ψ1 and ψ2 both normalized. Find a relationship between A and B such that the superposition Aψ1 + Bψ2 is also a normalized solution. I'm having trouble with the integral of |Aψ1 + Bψ2|2 dx.   Thank you!

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