1. Home
  2. University physics volume 3
  3. Unit 2. modern physics
  4. Quantum mechanics
  5. The quantum particle in a box

7.4 The quantum particle in a box  (Page 4/12)

<< Chapter < Page
  University physics volume 3     Page 4 / 12
Chapter >> Page >
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 ℏ 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.

Got questions? Get instant answers now!

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%

Got questions? Get instant answers now!

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 / ℏ sin n π x L , n = 1 , 2 , 3 , . . .

Questions & Answers

summarize halerambos & holbon
David Reply
the Three stages of Auguste Comte
Clementina Reply
what are agents of socialization
Antonio Reply
sociology of education
Nuhu Reply
definition of sociology of education
Nuhu
what is culture
Abdulrahim Reply
shared beliefs, values, and practices
AI-Robot
What are the two type of scientific method
ogunniran Reply
I'm willing to join you
Aceng Reply
what are the scientific method of sociology
Man
what is socialization
ogunniran Reply
the process wherein people come to understand societal norms and expectations, to accept society's beliefs, and to be aware of societal values
AI-Robot
scientific method in doing research
ogunniran
defimition of sickness in afica
Anita
Cosmology
ogunniran
Hmmm
ogunniran
list and explain the terms that found in society
REMMY Reply
list and explain the terms that found in society
Mukhtar
what are the agents of socialization
Antonio
Family Peer group Institution
Abdulwajud
I mean the definition
Antonio
ways of perceived deviance indifferent society
Naomi Reply
reasons of joining groups
SAM
to bring development to the nation at large
Hyellafiya
entails of consultative and consensus building from others
Gadama
World first Sociologist?
Abu
What is evolutionary model
Muhammad Reply
Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u
faruk
what are the modern trends in religious behaviours
Selekeye Reply
what are social norms
Daniel Reply
shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society
Lucius
that is how i understood it
Lucius
examples of societal norms
Diamond
Discuss the characteristics of the research located within positivist and the interpretivist paradigm
Tariro Reply
what is Industrialisation
Selekeye Reply
industrialization
Angelo
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 7

Read also:

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