Chapter 5: Bound States: Simple Cases

To describe the matter wave, does the function $\mathit{A}\mathit{s}\mathit{i}\mathit{n}\left(kx\right)\mathit{c}\mathit{o}\mathit{s}\mathit{\omega }\mathit{t}$have well-defined energy? Explain

does the wave function have a well-defined $\mathit{\psi }\left(x\right)\mathbf{=}\mathit{A}\left(e^{ikx}+e^{-ikx}\right)$momentum? Explain.

When is the temporal part of the wave function $0$? Why is this important?

Question: the operator for angular momentum about the z-axis in spherical polar coordinate is $-i\hslash \left(\frac{\partial }{\partial \varphi }\right)$.find the function $f\left(\varphi \right)$ that would have a well-defined z-component of angular momentum.

Show that$\mathit{\Delta }\mathit{p}\mathbf{=}\mathbf{0}\mathbf{\Rightarrow }\widehat{\mathbf{p}}\mathit{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\overline{\mathbf{p}}\mathit{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ that is, verify that unless the wave function is an Eigen function of the momentum operator, there will be a nonzero uncertainty in the momentumstarts with showing that the quantity

$\underset{\mathbf{a}\mathbf{l}\mathbf{l}\mathbf{s}\mathbf{p}\mathbf{a}\mathbf{c}\mathbf{e}}{\mathbf{\int }}{\mathit{\psi }}^{\mathbf{\ast }}\mathbf{\left(}\mathbf{x}\mathbf{\right)}{\mathbf{(}\widehat{\mathbf{p}}\mathbf{-}\overline{\mathbf{p}}\mathbf{)}}^{\mathbf{2}}\mathit{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{d}\mathit{x}$

Is ${\mathbf{\left(}\mathbf{\Delta }\mathbf{p}\mathbf{\right)}}^{\mathbf{2}}$. Then using the differential operator form of$\widehat{\mathbf{p}}$and integration by parts, show that it is also,

$\underset{\mathbf{a}\mathbf{l}\mathbf{l}\mathbf{s}\mathbf{p}\mathbf{a}\mathbf{c}\mathbf{e}}{\mathbf{\int }}{\mathbf{\left\{}\mathbf{(}\widehat{\mathbf{p}}\mathbf{-}\overline{\mathbf{p}}\mathbf{)}\mathbf{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right\}}}^{\mathbf{\ast }}\mathbf{\left\{}\mathbf{(}\widehat{\mathbf{p}}\mathbf{-}\overline{\mathbf{p}}\mathbf{)}\mathbf{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right\}}\mathit{d}\mathit{x}$

Together these show that if$\mathit{\Delta }\mathit{p}$is. 0. then the preceding quantity must be 0. However, the Integral of the complex square of a function(the quantity in the brackets) can only be 0 if the function is identically 0, so the assertion is proved.

In a study of heat transfer, we find that for a solid rod, there is a relationship between the second derivative of the temperature with respect to position along the rod and the first with respect to time. (A linear temperature change with position would imply as much heat flowing into a region as out. so the temperature there would not change with time).

$\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{T}\left(x,\tau \right)}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{2}}}\mathbf{=}\mathbf{\beta }\frac{\mathbf{\partial }\mathbf{T}\left(x,\tau \right)}{\mathbf{\partial }\mathbf{\tau }}\mathbf{\delta x}$

(a) Separate variables this assume a solution that is a product of a function of xand a function of tplug it in then divide by it, obtain two ordinary differential equations.

(b) consider a fairly simple, if somewhat unrealistic case suppose the temperature is 0 atx=0and, and x=1 positive in between, write down the simplest function of xthat (1) fits these conditions and (2) obey the differential equation involving x.Does your choice determine the value, including sign of some constant ?

(c) Obtain the full$\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}$for this case.

We learned that to be normalizable, a wave function must not itself diverge and must fall to 0 faster than ${\left|x\right|}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}$as x get large. Nevertheless. We find two functions that slightly violate these requirements very useful. Consider the quantum mechanical plane wave Ae^i(kx-ax)and the weird function${\mathbf{\Psi }}_{{\mathbf{x}}_{\mathbf{0}}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$pictured which we here call by its proper name. the Dirac delta function.

(a) Which of the two normalizability requirements is violated by the plane wave, and which by Dirac delta function?

(b) Normalization of the plane wave could be accomplished if it were simply truncated, restricted to the region -b<x<+bbeing identically 0 outside. What would then be the relationship between b and A, and what would happen to A as b approaches infinity?

(c) Rather than an infinitely tall and narrow spike like the Dirac delta function. Consider a function that is 0 everywhere except the narrow region$\mathbf{-}\mathbf{E}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{+}\mathbf{\partial }$ where its value is a constant B. This too could be normalized, What would be the relationship between s and B, and what would happen to B as s approaches 0? (What we get is not exactly the Dirac delta function, but the distinction involves comparing infinities, a dangerous business that we will avoid.)

(d) As we see, the two "exceptional" functions may be viewed as limits of normalizable ones. In those limits, they are also complementary to each other in terms of their position and momentum uncertainties. Without getting into calculations, describe how they are complementary.

The figure shows a potential energy function.

(a) How much energy could a classical particle have and still be bound?

(b) Where would an unbound particle have its maximum kinetic energy?

(c) For what range of energies might a classical particle be bound in either of two different regions?

(d) Do you think that a quantum mechanical particle with energy in the range referred to in part?

(e) Would be bound in one region or the other? Explain.

Consider a particle of mass mand energy E in a region where the potential energy is constant U₀. Greater than E and the region extends to$\mathbf{x}\mathbf{=}\mathbf{+}\mathbf{\infty }$

(a) Guess a physically acceptable solution of the Schrodinger equation in this region and demonstrate that it is solution,

(b) The region noted in part extends from x = + 1 nm to $+\infty$. To the left of x = 1nm. The particle’s wave function is Dcos (10⁹m^-1 x). Is also greater than Ehere?

(c) The particle’s mass m is 10^-3 kg. By how much (in eV) doesthe potential energy prevailing from x=1 nm to U₀. Exceed the particle’s energy?

Outline a procedure for predicting how the quantum-mechanically allowed energies for a harmonic oscillator should depend on a quantum number. In essence, allowed kinetic energies are the particle-in-a box energies, except the length Lis replaced by the distance between classical tuning points. Expressed in terms of E. Apply this procedure to a potential energy of the form U(x) = - b/x where b is a constant. Assume that at the origin there is an infinitely high wall, making it one turning point, and determine the other numing point in terms of E. For the average potential energy, use its value at half way between the tuning points. Again in terms of E. Find and expression for the allowed energies in terms of m, b, and n. (Although three dimensional, the hydrogen atom potential energy is of this form. and the allowed energy levels depend on a quantum number exactly as this simple model predicts.)