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Chapter 5: Q32E (page 188)

A finite potential energy function U(x) allows ψ(x) the solution of the time-independent Schrödinger equation. to penetrate the classically forbidden region. Without assuming any particular function for U(x) show that b(x) must have an inflection point at any value of x where it enters a classically forbidden region.

Short Answer

Expert verified

The answer of given problem isψx=Cexp+αxx<0Asin(kL) + Bcos(kL)0⩽x⩽LGexp-αxx>L

Step by step solution

01

Wave function

The wave function ψx

role="math" localid="1660033358067" ψx=Cexp+αxx<0Asin(kL) + Bcos(kL)0⩽x⩽LGexp-αxx>L

02

Step 2:Total energy and potential energy

By definition, the inflection point is a point where the total energy is equal to zero. Total energy E is equal to potential energy Uo

Traditionally authorised and classically banned territories are separated by turning points. A turning point is a point at which the second derivative vanishes.

So, the answer isψx=Cexp+αxx<0Asin(kL) + Bcos(kL)0⩽x⩽LGexp-αxx>L

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Most popular questions from this chapter

A 2kg block oscillates with an amplitude of 10cm on a spring of force constant 120 N/m .

(a) In which quantum state is the block?

(b) The block has a slight electric charge and drops to a lower energy level by generating a photon. What is the minimum energy decrease possible, and what would be the corresponding fractional change in energy?

Sketch the wave function. Is it smooth?

ψ(x)={2a3xe-axX>00X<0

To a good approximation. the hydrogen chloride molecule, HCI, behaves vibrationally as a quantum harmonic ascillator of spring constant 480N/mand with effective osciltating mass just that of the lighter atom, hydrogen If it were in its ground vibtational state, what wave. Iength photon would be just right to bump this molecule. up to its next-higher vibrational energy state?.

Equation 5-16 gives infinite well energies. Because equation 5-22 cannot be solved in closed form, there is no similar compact formula for finite well energies. Still many conclusions can be drawn without one. Argue on largely qualitative grounds that if the walls of a finite well are moved close together but not changed in height, then the well must progressively hold fewer bound states. (Make a clear connection between the width of the well and the height of the walls).

Consider the wave function that is a combination of two different infinite well stationary states the nth and the mth

ψx,t=12ψnxe-iEn/ℏt+12ψme-iEm/ℏt

  1. Show that the ψx,tis properly normalized.
  2. Show that the expectation value of the energy is the average of the two energies:E¯=12En+Em
  3. Show that the expectation value of the square of the energy is given by .
  4. Determine the uncertainty in the energy.
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