Physics for Scientists and Engineers, Vol. 1
Physics for Scientists and Engineers, Vol. 1
6th Edition
ISBN: 9781429201322
Author: Paul A. Tipler, Gene Mosca
Publisher: Macmillan Higher Education
Question
Book Icon
Chapter 34, Problem 51P

(a)

To determine

The value of xandx2 for a particle in nth state.

(a)

Expert Solution
Check Mark

Answer to Problem 51P

The value of x is 12L and value of x2 is 13L2L22n2π2 for a particle in nth state.

Explanation of Solution

Given:

The one-dimensional box region is 0xL .

The particle is in the nth state.

The wave functionis ψn(x)=2LsinnπxLfor,n=1,2,3,... .

Formula used:

The expression for x in nth state is given by,

  x=x( ψ n ( x ))2dx

The expression for x2 in nth state is given by,

  x2=x2( ψ n ( x ))2dx

The integral formula,

  θsin2θdθ=θ24θsin2θ4cos2θ8+c

The integral formula,

  θ2sin2θdθ=θ36(θ2418)sin2θθcos2θ4+c

Calculation:

Let, nπxL=θ .

By differentiating both sides,

  nπdxL=dθdx=Ldθnπ

The limit is 0toL changes to 0tonπ .

The x is calculated as,

  x=x ( ψ n ( x ) )2dx=0Lx( 2 L sin nπx L )2dx=0nπ( Lθ nπ)2Lsin2θ(Ldθnπ)=(2Ln2π2)0nπθsin2θdθ

Solving further as,

  x=( 2L n 2 π 2 )0nπθ sin2θdθ=(2L n 2 π 2)[ θ 24θsin2θ4cos2θ8]0nπ=(2L n 2 π 2)[n2π2400]=L2

The x2 is calculated as,

  x2=x2 ( ψ n ( x ) )2dx=0Lx2( 2 L sin nπx L )2dx=0nπ( Lθ nπ )22Lsin2θ(Ldθnπ)=(2L2n3π3)0nπθsin2θdθ

Solving further as,

  x2=( 2 L 2 n 3 π 3 )0nπθ2 sin2θdθ=(2 L 2 n 3 π 3)[ θ 36( θ 2 4 1 8)sin2θθcos2θ4]0nπ=(2 L 2 n 3 π 3)[n3π360nπ4]=13L2L22n2π2

Conclusion:

Therefore, the value of x is 12L and value of x2 is 13L2L22n2π2 for a particle in nth state.

(b)

To determine

The comparison of xandx2 for n1 with the value of problem 50 .

(b)

Expert Solution
Check Mark

Answer to Problem 51P

The value of xandx2 for the classical probability distribution function, P(x)=1L and for the wave function ψn(x)=2LsinnπxL in the nth state while n1 is same.

Explanation of Solution

Calculation:

The value of x in both the cases is same and equalto L2 .

The value of x2 for n1 is calculated as,

  x2=13L2L22n2π2=13L20=13L2

Which is same as in case of problem 50 .

Conclusion:

Therefore, the value of xandx2 for the classical probability distribution function, P(x)=1L and for the wave function ψn(x)=2LsinnπxL in the nth state while n1issame.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
Suppose there are two transformers between your house and the high-voltage transmission line that distributes the power. In addition, assume your house is the only one using electric power. At a substation the primary of a step-down transformer (turns ratio = 1:23) receives the voltage from the high-voltage transmission line. Because of your usage, a current of 51.1 mA exists in the primary of the transformer. The secondary is connected to the primary of another step- down transformer (turns ratio = 1:36) somewhere near your house, perhaps up on a telephone pole. The secondary of this transformer delivers a 240-V emf to your house. How much power is your house using? Remember that the current and voltage given in this problem are rms values.
In some places, insect "zappers," with their blue lights, are a familiar sight on a summer's night. These devices use a high voltage to electrocute insects. One such device uses an ac voltage of 3970 V, which is obtained from a standard 120-V outlet by means of a transformer. If the primary coil has 27 turns, how many turns are in the secondary coil? hel lp?
Hi,  Does Quantum physics theory means all branches for example quantum relativity, Quantum mechanics, Quantum field theory, and string theory? Can you explain each one of them? Best
Knowledge Booster
Background pattern image
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning
Text book image
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Text book image
University Physics Volume 3
Physics
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:OpenStax
Text book image
Physics for Scientists and Engineers with Modern ...
Physics
ISBN:9781337553292
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Text book image
Modern Physics
Physics
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Cengage Learning
Text book image
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning