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For a total energy of 0, the potential energy is given in Exercise 96. (a) Given these, to what region of the x-axis would a classical particle be restricted? Is the quantum-mechanical particle similarly restricted? (b) Write an expression for the probability that the (quantum-mechanical) particle would be found in the classically forbidden region, leaving it in the form of an integral. (The integral cannot be evaluated in closed form.)

Short Answer

Expert verified

(a) Classically allowed region is -3b,3b.

(b) Forbidden region is 230bA2X2e-x2ib2dx.

Step by step solution

01

Relationship between the total energy and the potential energy

The half of the potential energy is known as the total energy.

02

Step 2(a): The classically allowed region

As we know the potential energy is

Ux=h22mb4x2-3h22mb2Ux=22mb4x2-3×22mb2=22m-3b2+1b4x2

The potential energy is smaller than 0 when the region is classically allowed.

-3b2+1b4x2=0x2=3b2x=-3b,3b

So, the classically allowed region is -3b,3b.

The wave function extends infinitely far in both directions, so the quantum entity is not restricted to this region.

03

The probability of the particle

The probability for the particle is

-3b3bΨxΨ0xdx=-3b3bA2x2e-x2ib2dx=203bA2x2e-x2ib2dx

Therefore, the probability that the particle would be found in the classical forbidden region is 203bA2x2e-x2ib2dx.

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Most popular questions from this chapter

For the harmonic oscillator potential energy, U=12kx2, the ground-state wave function is ψ(x)=Ae-(mk/2)x2, and its energy is 12k/m.

(a) Find the classical turning points for a particle with this energy.

(b) The Schrödinger equation says that ψ(x) and its second derivative should be of the opposite sign when E > Uand of the same sign when E < U . These two regions are divided by the classical turning points. Verify the relationship between ψ(x)and its second derivative for the ground-state oscillator wave function.

(Hint:Look for the inflection points.)

The deeper the finite well, the more state it holds. In fact, a new state, the, is added when the well’s depthU0reachesh2(n1)2/8mL2. (a) Argue that this should be the case based only onk=2mE/h2, the shape of the wave inside, and the degree of penetration of the classically forbidden region expected for a state whose energy E is only negligibly belowU0. (b) How many states would be found up to this same “height” in an infinite well.

Given that the particle’s total energy is0, show that the potential energy is role="math" localid="1657529957489" U(x)=h22mb4x2-3h22mb2.

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.

What is the product of uncertainties determined in Exercise 60 and 61? Explain.

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