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The probability distributions Psi amplitude squared for the n=1 state, for the n=2 state, and for the n=20 are plotted as functions of x from x=0 to x=L. Psi sub 1 squared is maximum in the middle of the box, decreases to either side and goes to zero at the ends. Psi sub 2 squared is zero value in the middle of the box and at the ends, and has two equal value maxima. Psi sub 20 squared has twenty maxima, all of the same size, and goes to zero between them and at the ends.
The probability density distribution | ψ n ( x ) | 2 for a quantum particle in a box for: (a) the ground state, n = 1 ; (b) the first excited state, n = 2 ; and, (c) the nineteenth excited state, n = 20 .

The probability density of finding a classical particle between x and x + Δ x depends on how much time Δ t the particle spends in this region. Assuming that its speed u is constant, this time is Δ t = Δ x / u , which is also constant for any location between the walls. Therefore, the probability density of finding the classical particle at x is uniform throughout the box, and there is no preferable location for finding a classical particle. This classical picture is matched in the limit of large quantum numbers. For example, when a quantum particle is in a highly excited state, shown in [link] , the probability density is characterized by rapid fluctuations and then the probability of finding the quantum particle in the interval Δ x does not depend on where this interval is located between the walls.

A classical particle in a box

A small 0.40-kg cart is moving back and forth along an air track between two bumpers located 2.0 m apart. We assume no friction; collisions with the bumpers are perfectly elastic so that between the bumpers, the car maintains a constant speed of 0.50 m/s. Treating the cart as a quantum particle, estimate the value of the principal quantum number that corresponds to its classical energy.

Strategy

We find the kinetic energy K of the cart and its ground state energy E 1 as though it were a quantum particle. The energy of the cart is completely kinetic, so K = n 2 E 1 ( [link] ). Solving for n gives n = ( K / E 1 ) 1 / 2 .

Solution

The kinetic energy of the cart is

K = 1 2 m u 2 = 1 2 ( 0.40 kg ) ( 0.50 m/s ) 2 = 0.050 J .

The ground state of the cart, treated as a quantum particle, is

E 1 = π 2 2 2 m L 2 = π 2 ( 1.05 × 10 −34 J · s ) 2 2 ( 0.40 kg ) ( 2.0 m ) 2 = 1.700 × 10 −68 J .

Therefore, n = ( K / E 1 ) 1 / 2 = ( 0.050 / 1.700 × 10 −68 ) 1 / 2 = 1.2 × 10 33 .

Significance

We see from this example that the energy of a classical system is characterized by a very large quantum number. Bohr’s correspondence principle    concerns this kind of situation. We can apply the formalism of quantum mechanics to any kind of system, quantum or classical, and the results are correct in each case. In the limit of high quantum numbers, there is no advantage in using quantum formalism because we can obtain the same results with the less complicated formalism of classical mechanics. However, we cannot apply classical formalism to a quantum system in a low-number energy state.

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Check Your Understanding (a) Consider an infinite square well with wall boundaries x = 0 and x = L . What is the probability of finding a quantum particle in its ground state somewhere between x = 0 and x = L / 4 ? (b) Repeat question (a) for a classical particle.

a. 9.1%; b. 25%

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Having found the stationary states ψ n ( x ) and the energies E n by solving the time-independent Schrӧdinger equation [link] , we use [link] to write wave functions Ψ n ( x , t ) that are solutions of the time-dependent Schrӧdinger’s equation given by [link] . For a particle in a box this gives

Ψ n ( x , t ) = e i ω n t ψ n ( x ) = 2 L e i E n t / sin n π x L , n = 1 , 2 , 3 , . . .

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

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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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