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By the end of this section, you will be able to:
  • Describe how to set up a boundary-value problem for the stationary Schrӧdinger equation
  • Explain why the energy of a quantum particle in a box is quantized
  • Describe the physical meaning of stationary solutions to Schrӧdinger’s equation and the connection of these solutions with time-dependent quantum states
  • Explain the physical meaning of Bohr’s correspondence principle

In this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energy of the particle is quantized as a consequence of a standing wave condition inside the box.

Consider a particle of mass m that is allowed to move only along the x -direction and its motion is confined to the region between hard and rigid walls located at x = 0 and at x = L ( [link] ). Between the walls, the particle moves freely. This physical situation is called the infinite square well    , described by the potential energy function

U ( x ) = { 0 , 0 x L , , otherwise .

Combining this equation with Schrӧdinger’s time-independent wave equation gives

2 2 m d 2 ψ ( x ) d x 2 = E ψ ( x ) , for 0 x L

where E is the total energy of the particle. What types of solutions do we expect? The energy of the particle is a positive number, so if the value of the wave function is positive (right side of the equation), the curvature of the wave function is negative, or concave down (left side of the equation). Similarly, if the value of the wave function is negative (right side of the equation), the curvature of the wave function is positive or concave up (left side of equation). This condition is met by an oscillating wave function, such as a sine or cosine wave. Since these waves are confined to the box, we envision standing waves with fixed endpoints at x = 0 and x = L .

The potential U is plotted as a function of x. U is equal to infinity at x equal to or less than zero, and at x equal to or greater than L. U is equal to zero between x = 0 and x = L.
The potential energy function that confines the particle in a one-dimensional box.

Solutions ψ ( x ) to this equation have a probabilistic interpretation. In particular, the square | ψ ( x ) | 2 represents the probability density of finding the particle at a particular location x . This function must be integrated to determine the probability of finding the particle in some interval of space. We are therefore looking for a normalizable solution that satisfies the following normalization condition:

0 L d x | ψ ( x ) | 2 = 1 .

The walls are rigid and impenetrable, which means that the particle is never found beyond the wall. Mathematically, this means that the solution must vanish at the walls:

ψ ( 0 ) = ψ ( L ) = 0 .

We expect oscillating solutions, so the most general solution to this equation is

ψ k ( x ) = A k cos k x + B k sin k x

where k is the wave number, and A k and B k are constants. Applying the boundary condition expressed by [link] gives

ψ k ( 0 ) = A k cos ( k · 0 ) + B k sin ( k · 0 ) = A k = 0 .

Because we have A k = 0 , the solution must be

ψ k ( x ) = B k sin k x .

If B k is zero, ψ k ( x ) = 0 for all values of x and the normalization condition, [link] , cannot be satisfied. Assuming B k 0 , [link] for x = L then gives

0 = B k sin ( k L ) sin ( k L ) = 0 k L = n π , n = 1 , 2 , 3 ,...

We discard the n = 0 solution because ψ ( x ) for this quantum number would be zero everywhere—an un-normalizable and therefore unphysical solution. Substituting [link] into [link] gives

Questions & Answers

A Pb wire wound in a tight solenoid of diameter of 4.0 mm is cooled to a temperature of 5.0 K. The wire is connected in series with a 50-Ωresistor and a variable source of emf. As the emf is increased, what value does it have when the superconductivity of the wire is destroyed?
Rupal Reply
how does colour appear in thin films
Nwjwr Reply
in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
Kimani Reply
derivation of lateral shieft
James Reply
hi
Imran
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
 Cavendish Experiment to Measure Gravitational Constant. ... This experiment used a torsion balance device to attract lead balls together, measuring the torque on a wire and equating it to the gravitational force between the balls. Then by a complex derivation, the value of G was determined.
Triio
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
acceleration also increases
Jay
there are two correct answers depending on whether air resistance is considered. none of those answers have acceleration increasing.
Michael
Acceleration is the change in velocity over time, hence it's the derivative of the velocity with respect to time. So this case would depend on the velocity. More specifically the change in velocity in the system.
Big
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
Practice Key Terms 7

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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