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Chapter 6: Q48P (page 1356)

Consider a particle in a box with rigid walls at x= 0 and x= L. Let the particle be in the ground level. Calculate the probability |蠄x|2dxthat the particle will be found in the interval x to x+ dxfor (a) x= L/4; (b) x= L/2; (c) x= 3L/4.

Short Answer

Expert verified
  1. The probability that the particle will be found in the interval x to x+ dxfor x= L/4 is dxL
  2. The probability that the particle will be found in the interval x to x+ dxfor x= L/2 is 2dxL
  3. The probability that the particle will be found in the interval x to x+ dxfor x= 3L/4 is dxL

Step by step solution

01

(a) Determination of the probability that the particle will be found in the interval x to x + dx for x = L/4.

The probability distribution between two points is given as,

鈭�x1x2蠄2dx

Also, the ground state wave function for particle in a box is given as,

蠄1=2LsinTTXL

Thus, the wave function at x = L/4,

蠄2dx=2Lsin2TTLL4dx=2Lsin2TT4dx=dxL

02

(b) Determination of the probability that the particle will be found in the interval x to x + dx for x = L/2.

Similarly, repeat the above calculation for x = L/2, the wave function is,

蠄2dx=2Lsin2TTLL2dx=2Lsin2TT2dx=2dxL

03

(c) Determination of the probability that the particle will be found in the interval x to x + dx for x = 3L/4.

Similarly, repeat the above calculation for x = 3L/4, the wave function is,

蠄2dx=2Lsin2TTL3L2dx=2Lsin23TT4dx=dxL

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