(a)
The expectation value of
(a)
Answer to Problem 57P
The expectation value of
Explanation of Solution
Write the equation for the expectation value of
Here,
Write the expression of the given wave function.
Write the expression for the complex conjugate of the given wave function.
Put equations (II) and (III) in equation (I) and rearrange it.
Integrate the above equation.
Conclusion:
Therefore, the expectation value of
(b)
The probability of finding the particle near
(b)
Answer to Problem 57P
The probability of finding the particle near
Explanation of Solution
Write the equation for the probability that the particle lies in the range
Here,
Put equations (II) and (III) in equation (IV) and rearrange it.
Integrate the above equation.
Conclusion:
Therefore, the probability of finding the particle near
(c)
The probability of finding the particle near
(c)
Answer to Problem 57P
The probability of finding the particle near
Explanation of Solution
Write the equation for the probability that the particle lies in the range
Put equations (II) and (III) in equation (IV) and rearrange it.
Integrate the above equation.
Conclusion:
Therefore, the probability of finding the particle near
(d)
The argument for the statement that the result of part (a) does not contradict the result of part (b) and part (c).
(d)
Answer to Problem 57P
It is more probable to find the particle either near
Explanation of Solution
Probability density is the relative probability per unit volume that the particle will be found at any given point in the volume. The probability density for the given function with
The expectation value of
Conclusion:
Thus, it is more probable to find the particle either near
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Chapter 28 Solutions
Principles of Physics: A Calculus-Based Text
- A quantum particle in an infinitely deep square well has a wave function that is given by ψ1(x) = √2/Lsin (πx/L)for 0 ≤ x ≤ L and is zero otherwise. (a) Determine the probability of finding the particle between x = 0 and x = 1/3L.(b) Use the result of this calculation and a symmetry argumentto find the probability of finding the particle between x = 1/3 L and x = 2/3 L. Do not re-evaluate the integral.arrow_forwardFor a "particle in a box" of length, L, the wavelength for the nth level is given by An 2L %3D 2п and the wave function is n(x) = A sin (x) = A sin (x). The energy levels are пп %3D n?h? given by En : %3D 8mL2 lPn(x)|2 is the probability of finding the particle at position x in the box. Since the particle must be somewhere in the box, the integral of this function over the length of the box must be equal to 1. This is the normalization condition and ensuring that this is the case is called “normalizing" the wave function. Find the value of A the amplitude of the wave function, that normalizes it. Write the normalized wave function for the nth state of the particle in a box.arrow_forward(a) Find the normalization constant A for a wave function made up of the two lowest states of a quantum particle in a box extending from x= 0 to x = L: x) = A sin + 4 sin L. (b) A particle is described in the space -aSxs a by the wave function (x) = A cos + B sin 2a a Determine the relationship between the values of A and B required for normalization.arrow_forward
- A hypothetical one dimensional quantum particle has a normalised wave function given by (x) = ax - iß, where a and 3 are real constants and i = √-1. What is the most likely. x-position, II(x), for the particle to be found at? 0 11(x) == ○ II(2) = 0 ○ II(r) = 2/ O II(z) = 011(r) = ± √ +√ 13 aarrow_forwardFor a quantum particle described by a wave function (x), the expectation value of a physical quantity f(x) associated with the particle is defined by For a particle in an infinitely deep one-dimensional box extending from x = 0 to x = L, show that L 2n°7arrow_forwardWhat is the probability of the particle that in the box with a length of 2 nm is between x = 0.2 and x = 1.0 nm? Ѱ=√(2/L)*sin(nπx/L)arrow_forward
- A quantum particle is described by the wave function ψ(x) = A cos (2πx/L) for −L/4 ≤x ≤ L/4 and ψ(x) everywhere else. Determine:(a) The normalization constant A,(b) The probability of finding the electron between x = 0 and x = L/8.arrow_forwardA quantum system is described by a wave function (r) being a superposition of two states with different energies E1 and E2: (x) = c191(r)e iEit/h+ c292(x)e¯iE2t/h. where ci = 2icz and the real functions p1(x) and p2(r) have the following properties: vile)dz = ile)dz = 1, "0 = rp(x)T#(x)l& p1(x)92(x)dx% D0. Calculate: 1. Probabilities of measurement of energies E1 and E2 2. Expectation valuc of cnergy (E)arrow_forwardFor a particle in a one-dimensional box, calculate the probability of the particle to exists between the length of 0.30L and 0.70L if n = 5.arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning