<< Chapter < Page Chapter >> Page >

Expectation value (part ii)

The time-dependent wave function of a particle confined to a region between 0 and L is

ψ ( x , t ) = A e i ω t sin ( π x / L )

where ω is angular frequency and E is the energy of the particle. ( Note: The function varies as a sine because of the limits (0 to L ). When x = 0 , the sine factor is zero and the wave function is zero, consistent with the boundary conditions.) Calculate the expectation values of position, momentum, and kinetic energy.

Strategy

We must first normalize the wave function to find A . Then we use the operators to calculate the expectation values.

Solution

Computation of the normalization constant:

1 = 0 L d x ψ * ( x ) ψ ( x ) = 0 L d x ( A e + i ω t sin π x L ) ( A e i ω t sin π x L ) = A 2 0 L d x sin 2 π x L = A 2 L 2 A = 2 L .

The expectation value of position is

x = 0 L d x ψ * ( x ) x ψ ( x ) = 0 L d x ( A e + i ω t sin π x L ) x ( A e i ω t sin π x L ) = A 2 0 L d x x sin 2 π x L = A 2 L 2 4 = L 2 .

The expectation value of momentum in the x -direction also requires an integral. To set this integral up, the associated operator must— by rule—act to the right on the wave function ψ ( x ) :

i d d x ψ ( x ) = i d d x A e i ω t sin π x L = i A h 2 L e i ω t cos π x L .

Therefore, the expectation value of momentum is

p = 0 L d x ( A e + i ω t sin π x L ) ( i A h 2 L e i ω t cos π x L ) = i A 2 h 4 L 0 L d x sin 2 π x L = 0 .

The function in the integral is a sine function with a wavelength equal to the width of the well, L —an odd function about x = L / 2 . As a result, the integral vanishes.

The expectation value of kinetic energy in the x -direction requires the associated operator to act on the wave function:

2 2 m d 2 d x 2 ψ ( x ) = 2 2 m d 2 d x 2 A e i ω t sin π x L = 2 2 m A e i ω t d 2 d x 2 sin π x L = A h 2 2 m L 2 e i ω t sin π x L .

Thus, the expectation value of the kinetic energy is

K = 0 L d x ( A e + i ω t sin π x L ) ( A h 2 2 m L 2 e i ω t sin π x L ) = A 2 h 2 2 m L 2 0 L d x sin 2 π x L = A 2 h 2 2 m L 2 L 2 = h 2 2 m L 2 .

Significance

The average position of a large number of particles in this state is L /2. The average momentum of these particles is zero because a given particle is equally likely to be moving right or left. However, the particle is not at rest because its average kinetic energy is not zero. Finally, the probability density is

| ψ | 2 = ( 2 / L ) sin 2 ( π x / L ) .

This probability density is largest at location L /2 and is zero at x = 0 and at x = L . Note that these conclusions do not depend explicitly on time.

Check Your Understanding For the particle in the above example, find the probability of locating it between positions 0 and L /4

( 1 / 2 1 / π ) / 2 = 9 %

Got questions? Get instant answers now!

Quantum mechanics makes many surprising predictions. However, in 1920, Niels Bohr (founder of the Niels Bohr Institute in Copenhagen, from which we get the term “Copenhagen interpretation”) asserted that the predictions of quantum mechanics and classical mechanics must agree for all macroscopic systems, such as orbiting planets, bouncing balls, rocking chairs, and springs. This correspondence principle    is now generally accepted. It suggests the rules of classical mechanics are an approximation of the rules of quantum mechanics for systems with very large energies. Quantum mechanics describes both the microscopic and macroscopic world, but classical mechanics describes only the latter.

Summary

  • In quantum mechanics, the state of a physical system is represented by a wave function.
  • In Born’s interpretation, the square of the particle’s wave function represents the probability density of finding the particle around a specific location in space.
  • Wave functions must first be normalized before using them to make predictions.
  • The expectation value is the average value of a quantity that requires a wave function and an integration.

Conceptual questions

What is the physical unit of a wave function, Ψ ( x , t ) ? What is the physical unit of the square of this wave function?

1 / L , where L = length ; 1/ L , where L = length

Got questions? Get instant answers now!

Can the magnitude of a wave function ( Ψ * ( x , t ) Ψ ( x , t ) ) be a negative number? Explain.

Got questions? Get instant answers now!

What kind of physical quantity does a wave function of an electron represent?

The wave function does not correspond directly to any measured quantity. It is a tool for predicting the values of physical quantities.

Got questions? Get instant answers now!

What is the physical meaning of a wave function of a particle?

Got questions? Get instant answers now!

What is the meaning of the expression “expectation value?” Explain.

The average value of the physical quantity for a large number of particles with the same wave function.

Got questions? Get instant answers now!

Problems

Compute | Ψ ( x , t ) | 2 for the function Ψ ( x , t ) = ψ ( x ) sin ω t , where ω is a real constant.

| ψ ( x ) | 2 sin 2 ω t

Got questions? Get instant answers now!

Given the complex-valued function f ( x , y ) = ( x i y ) / ( x + i y ) , calculate | f ( x , y ) | 2 .

Got questions? Get instant answers now!

Which one of the following functions, and why, qualifies to be a wave function of a particle that can move along the entire real axis? (a) ψ ( x ) = A e x 2 ;
(b) ψ ( x ) = A e x ; (c) ψ ( x ) = A tan x ;
(d) ψ ( x ) = A ( sin x ) / x ; (e) ψ ( x ) = A e | x | .

(a) and (e), can be normalized

Got questions? Get instant answers now!

A particle with mass m moving along the x -axis and its quantum state is represented by the following wave function:

Ψ ( x , t ) = { 0 , x < 0 , A x e α x e i E t / , x 0 ,

where α = 2.0 × 10 10 m −1 . (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval 0 x L . (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.

Got questions? Get instant answers now!

A wave function of a particle with mass m is given by

ψ ( x ) = { A cos α x , π 2 α x + π 2 α , 0 , otherwise,

where α = 1.00 × 10 10 / m . (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval 0 x 0.5 × 10 −10 m . (c) Find the particle’s average position. (d) Find its average momentum. (e) Find its average kinetic energy −0.5 × 10 −10 m x + 0.5 × 10 −10 m .

a. A = 2 α / π ; b. probability = 29.3 % ; c. x = 0 ; d. p = 0 ; e. K = α 2 2 / 2 m

Got questions? Get instant answers now!

Questions & Answers

in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
Kimani Reply
derivation of lateral shieft
James Reply
Hi
Amjad
Hi
Amjad
hi
ALFRED
how are you?
Amjad
hi
asif
hi
Imran
I'm fine
ALFRED
total binding energy of ionic crystal at equilibrium is
All Reply
How does, ray of light coming form focus, behaves in concave mirror after refraction?
Bishesh Reply
Refraction does not occur in concave mirror. If refraction occurs then I don't know about this.
Sushant
What is motion
Izevbogie Reply
Anything which changes itself with respect to time or surrounding
Sushant
good
Chemist
and what's time? is time everywhere same
Chemist
No
Sushant
how can u say that
Chemist
do u know about black hole
Chemist
Not so more
Sushant
Radioactive substance
DHEERAJ
These substance create harmful radiation like alpha particle radiation, beta particle radiation, gamma particle radiation
Sushant
But ask anything changes itself with respect to time or surrounding A Not any harmful radiation
DHEERAJ
explain cavendish experiment to determine the value of gravitational concept.
Celine Reply
For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
Evan Reply
as a free falling object increases speed what is happening to the acceleration
Success Reply
of course g is constant
Alwielland
acceleration also inc
Usman
which paper will be subjective and which one objective
jay
normal distributiin of errors report
Dennis
normal distribution of errors
Dennis
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
Rafi
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
Prema
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
Prema
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
Prema
this msg is out of mistake. sorry friends​.
Prema
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
John
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
Yoblaze
physics is the study of non living things if we added it with biology it becomes biophysics and bio is the study of living things tell me please what is this?
tahreem
physics is the study of matter,energy and their interactions
Buvanes
all living things are matter
Buvanes
why rolling friction is less than sliding friction
tahreem
thanks buvanas
tahreem
is this a physics forum
Physics Reply
explain l-s coupling
Depk Reply

Get the best University physics vol... course in your pocket!





Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 3' conversation and receive update notifications?

Ask