<< Chapter < Page Chapter >> Page >
d V = 4 π r 2 d r .

The probability of finding the electron in the region r to r + d r (“at approximately r ”) is

P ( r ) d r = | ψ n 00 | 2 4 π r 2 d r .

Here P ( r ) is called the radial probability density function    (a probability per unit length). For an electron in the ground state of hydrogen, the probability of finding an electron in the region r to r + d r is

| ψ n 00 | 2 4 π r 2 d r = ( 4 / a 0 3 ) r 2 exp ( −2 r / a 0 ) d r ,

where a 0 = 0.5 angstroms. The radial probability density function P ( r ) is plotted in [link] . The area under the curve between any two radial positions, say r 1 and r 2 , gives the probability of finding the electron in that radial range. To find the most probable radial position, we set the first derivative of this function to zero ( d P / d r = 0 ) and solve for r . The most probable radial position is not equal to the average or expectation value of the radial position because | ψ n 00 | 2 is not symmetrical about its peak value.

A graph of the function P of r as a function of r is shown. It is zero at r = 0, rises to a maximum at r = a sub 0, then gradually decreases and goes asymptotically to zero at large r. The maximum is at the most probable radial position. The area of the region under the curve from r sub 1 to r sub 2 is shaded.
The radial probability density function for the ground state of hydrogen.

If the electron has orbital angular momentum ( l 0 ), then the wave functions representing the electron depend on the angles θ and ϕ ; that is, ψ n l m = ψ n l m ( r , θ , ϕ ). Atomic orbitals for three states with n = 2 and l = 1 are shown in [link] . An atomic orbital    is a region in space that encloses a certain percentage (usually 90%) of the electron probability. (Sometimes atomic orbitals are referred to as “clouds” of probability.) Notice that these distributions are pronounced in certain directions. This directionality is important to chemists when they analyze how atoms are bound together to form molecules.

This diagram illustrates the shapes of p orbitals. The orbitals are dumbbell shaped and oriented along the x, y, and z axes.
The probability density distributions for three states with n = 2 and l = 1 . The distributions are directed along the (a) x -axis, (b) y -axis, and (c) z -axis.

A slightly different representation of the wave function is given in [link] . In this case, light and dark regions indicate locations of relatively high and low probability, respectively. In contrast to the Bohr model of the hydrogen atom, the electron does not move around the proton nucleus in a well-defined path. Indeed, the uncertainty principle makes it impossible to know how the electron gets from one place to another.

The figure shows probability clouds for electrons in the n equals 1, 2 and 3, l equals 0, 1 and 2 states in a 3 by 3 grid. n=1, l=0 is a spherically symmetric distribution, brighter in the center and gradually fading with increasing radius, with no nodes. n=2, l=0 is a spherically symmetric distribution with a spherical, concentric node. The node appears as a black circle within the cloud. The cloud is brightest in the center, fading to black at the node, brightening again to outside the node (but not as bright as at the center of the cloud), then fading again at large r. n=2, l=1 has a planar node along the diameter of the cloud, appearing as a dark line across the distribution and indentations at the edges. The cloud is brightest near the center, above and below the node. n=3, l=0 is a spherically symmetric distribution with two spherical, concentric nodes. The nodes appear as concentric black circles within the cloud. The cloud is brightest in the center, fading to black at the first node, brightening again to a maximum brightness outside the node, fading to black at the second node brightening again, then fading again at large r. The local maxima (at the center, between the nodes, and outside the outer node) decrease in intensity. n=3, l=2 has both a concentric circular node and a planar node along the diameter, appearing as a circle in and line across the cloud. The cloud is brightest inside the circular node. A second local maximum brightness is seen within the lobes above and below the planar node. n=3, l=2 has two planar nodes, which appear as an X across the cloud. The quarters of the cloud thus defined are deeply indented at the edges, forming rounded lobes. The cloud is brightest near the center.
Probability clouds for the electron in the ground state and several excited states of hydrogen. The probability of finding the electron is indicated by the shade of color; the lighter the coloring, the greater the chance of finding the electron.

Summary

  • A hydrogen atom can be described in terms of its wave function, probability density, total energy, and orbital angular momentum.
  • The state of an electron in a hydrogen atom is specified by its quantum numbers ( n , l , m ).
  • In contrast to the Bohr model of the atom, the Schrödinger model makes predictions based on probability statements.
  • The quantum numbers of a hydrogen atom can be used to calculate important information about the atom.

Conceptual questions

Identify the physical significance of each of the quantum numbers of the hydrogen atom.

n (principal quantum number) total energy
l (orbital angular quantum number) total absolute magnitude of the orbital angular momentum
m (orbital angular projection quantum number) z -component of the orbital angular momentum

Got questions? Get instant answers now!

Questions & Answers

what is force
Afework Reply
The different examples for collision
Afework
What is polarization and there are type
Muhammed Reply
Polarization is the process of transforming unpolarized light into polarized light. types of polarization 1. linear polarization. 2. circular polarization. 3. elliptical polarization.
Eze
Describe what you would see when looking at a body whose temperature is increased from 1000 K to 1,000,000 K
Aishwarya Reply
how is tan ninety minus an angle equals to cot an angle?
Niicommey Reply
please I don't understand all about this things going on here
Jeremiah Reply
What is torque?
Matthew Reply
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study.
Teka
Torque refers to the rotational force. i.e Torque = Force × radius.
Arun
Torque is the rotational equivalent of force . Specifically, it is a force exerted at a distance from an object's axis of rotation. In the same way that a force applied to an object will cause it to move linearly, a torque applied to an object will cause it to rotate around a pivot point.
Teka
Torque is the rotational equivalence of force . So, a net torque will cause an object to rotate with an angular acceleration. Because all rotational motions have an axis of rotation, a torque must be defined about a rotational axis. A torque is a force applied to a point on an object about the axis
Teka
When a missle is shot from one spaceship towards another, it leaves the first at 0.950c and approaches the other at 0.750c. what is the relative velocity of the two shipd
Marifel Reply
how to convert:m^3/s^2 all divided by kg to cm^3/s^2
Thibaza Reply
Is there any proof of existence of luminiferious aether ?
Zero Reply
mass conversion of 58.73kg =mg
Proactive Reply
is Space time fabric real
Godawari Reply
What's the relationship between the work function and the cut off frequency in the diagram above?
frankline Reply
due to the upthrust weight of the object varise with force in which the body fall into the water pendincular with the reflection of light with it
Gift
n=I/r
Gift
can someone explain what is going on here
falanga
so some pretty easy physics questions bring em
falanga
what is meant by fluctuated
Olasukanmi Reply
If n=cv then how v=cn? and if n=c/v then how v=cn?
Natanim
convert feet to metre
Mbah Reply
what is electrolysis
Mbah
Electrolysis is the chemical decomposition of electrolyte either in molten state or solution to conduct electricity
Ayomide
class ninekasindhtextbookurdusave
Ayesha Reply
can someone help explain why v2/c2 is =1/2 Using The Lorentz Transformation For Time Spacecraft S′ is on its way to Alpha Centauri when Spacecraft S passes it at relative speed c /2. The captain of S′ sends a radio signal that lasts 1.2 s according to that ship’s clock. Use the Lorentz transformati
Jennifer
Practice Key Terms 5

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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