<< Chapter < Page Chapter >> Page >
2 2 m d 2 d x 2 ( B k sin ( k x ) ) = E ( B k sin ( k x ) ) .

Computing these derivatives leads to

E = E k = 2 k 2 2 m .

According to de Broglie, p = k , so this expression implies that the total energy is equal to the kinetic energy, consistent with our assumption that the “particle moves freely.” Combining the results of [link] and [link] gives

E n = n 2 π 2 2 2 m L 2 , n = 1 , 2 , 3 , . . .

Strange! A particle bound to a one-dimensional box can only have certain discrete (quantized) values of energy. Further, the particle cannot have a zero kinetic energy—it is impossible for a particle bound to a box to be “at rest.”

To evaluate the allowed wave functions that correspond to these energies, we must find the normalization constant B n . We impose the normalization condition [link] on the wave function

ψ n ( x ) = B n sin n π x / L
1 = 0 L d x | ψ n ( x ) | 2 = 0 L d x B n 2 sin 2 n π L x = B n 2 0 L d x sin 2 n π L x = B n 2 L 2 B n = 2 L .

Hence, the wave functions that correspond to the energy values given in [link] are

ψ n ( x ) = 2 L sin n π x L , n = 1 , 2 , 3 , . . .

For the lowest energy state or ground state energy    , we have

E 1 = π 2 2 2 m L 2 , ψ 1 ( x ) = 2 L sin ( π x L ) .

All other energy states can be expressed as

E n = n 2 E 1 , ψ n ( x ) = 2 L sin ( n π x L ) .

The index n is called the energy quantum number    or principal quantum number    . The state for n = 2 is the first excited state, the state for n = 3 is the second excited state, and so on. The first three quantum states (for n = 1 , 2 , and 3 ) of a particle in a box are shown in [link] .

The wave functions in [link] are sometimes referred to as the “states of definite energy.” Particles in these states are said to occupy energy levels    , which are represented by the horizontal lines in [link] . Energy levels are analogous to rungs of a ladder that the particle can “climb” as it gains or loses energy.

The wave functions in [link] are also called stationary state     s and standing wave state     s . These functions are “stationary,” because their probability density functions, | Ψ ( x , t ) | 2 , do not vary in time, and “standing waves” because their real and imaginary parts oscillate up and down like a standing wave—like a rope waving between two children on a playground. Stationary states are states of definite energy [ [link] ], but linear combinations of these states, such as ψ ( x ) = a ψ 1 + b ψ 2 (also solutions to Schrӧdinger’s equation) are states of mixed energy.

The first three quantum states of a quantum particle in a box for principal quantum numbers n=1, n=2, and n=3 are shown: Figure (a) shown the graphs of the standing wave solutions. The vertical axis is the wave function, with a separate origin for each state that is aligned with the energy scale of figure (b). The horizontal axis is x from just below 0 to just past L. Figure (b) shows the energy of each of the states on the vertical E sub n axis. All of the wave functions are zero for x less than 0 and x greater than L. The n=1 function is the first half wave of the wavelength 2 L sine function and its energy is pi squared times h squared divided by the quantity 2 m L squared. The n=2 function is the first full wave of the wavelength 2 L sine function and its energy is 4 pi squared times h squared divided by the quantity 2 m L squared. The n=3 function is the first one and a half waves of the wavelength 2 L sine function and its energy is 9 pi squared times h squared divided by the quantity 2 m L squared.
The first three quantum states of a quantum particle in a box for principal quantum numbers n = 1 , 2 , and 3 : (a) standing wave solutions and (b) allowed energy states.

Energy quantization is a consequence of the boundary conditions. If the particle is not confined to a box but wanders freely, the allowed energies are continuous. However, in this case, only certain energies ( E 1 , 4 E 1 , 9 E 1 , …) are allowed. The energy difference between adjacent energy levels is given by

Δ E n + 1 , n = E n + 1 E n = ( n + 1 ) 2 E 1 n 2 E 1 = ( 2 n + 1 ) E 1 .

Conservation of energy demands that if the energy of the system changes, the energy difference is carried in some other form of energy. For the special case of a charged particle confined to a small volume (for example, in an atom), energy changes are often carried away by photons. The frequencies of the emitted photons give us information about the energy differences (spacings) of the system and the volume of containment—the size of the “box” [see [link] ].

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
Practice Key Terms 7

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