Chapter 5: Q97CE (page 193)

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 $(- \sqrt{3} b, \sqrt{3} b)$ .

(b) Forbidden region is $2^{\sqrt{3}} \int_{0}^{b} A^{2} X^{2} e^{- x^{2} i b^{2}} d x$ .

Step by step solution

Relationship between the total energy and the potential energy

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

Step 2(a): The classically allowed region

As we know the potential energy is

$\begin{array}{rcl} U (x) & = & \frac{h^{2}}{2 m b^{4}} x^{2} - \frac{3 h^{2}}{2 m b^{2}} \\ U (x) & = & \frac{2}{2 m b^{4}} x^{2} - \frac{3 \times 2}{2 m b^{2}} \\ = & \frac{2}{2 m} (- \frac{3}{b^{2}} + \frac{1}{b^{4}} x^{2}) \end{array}$

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

$\begin{array}{rcl} (- \frac{3}{b^{2}} + \frac{1}{b^{4}} x^{2}) & = & 0 \\ x^{2} & = & 3 b^{2} \\ x & = & - \sqrt{3} b, \sqrt{3} b \end{array}$

So, the classically allowed region is $(- \sqrt{3} b, \sqrt{3} b)$ .

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

The probability of the particle

The probability for the particle is

$\begin{array}{rcl} \int_{- \sqrt{3} b}^{\sqrt{3} b} Ψ (x) Ψ^{0} (x) d x & = & \int_{- \sqrt{3} b}^{\sqrt{3} b} A^{2} x^{2} e^{- x^{2} i b^{2}} d x \\ = & 2 \int_{0}^{\sqrt{3} b} A^{2} x^{2} e^{- x^{2} i b^{2}} d x \end{array}$

Therefore, the probability that the particle would be found in the classical forbidden region is $\begin{array}{rcl} 2 \int_{0}^{\sqrt{3} b} A^{2} x^{2} e^{- x^{2} i b^{2}} d x \end{array}$ .

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Short Answer

Step by step solution

Relationship between the total energy and the potential energy

Step 2(a): The classically allowed region

The probability of the particle

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