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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.


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 .


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 .


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

A 10kg mass lift to a height of 24m and release. what is the total energy of the system
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what is the difference between a molecule and atom
Natanim Reply
Atoms are single neutral particles. Molecules are neutral particles made of two or more atoms bonded together.
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A body quadruples its momentum when its speed doubles.What was the initial speed in units of c.i.e..what was u/c ?
Lekshmi Reply
what is enthalpy?
prabir Reply
a thermodynamic quantity equivalent to the total heat content of a system
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What is the meaning of Nuclear Fission?
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عند قذف جسم إلى أعلى بسرعة إبتدائية فإنه سيصل إلى ارتفاع معين (أقصى ارتفاع) ثم يعود هابطاً نحو سطح الأرض .   إذا قُذِفَ جسم إلى أعلى ووجد أن سرعته 18 م / ث عندما قطع 1/4 المسافة التي تمثل أقصى ارتفاع سيصله فالمطلوب إيجاد السرعة التي قُذِف بها بالمتر / ث . إن هذه السرعة هي واحدة من الإجابات التالية
Aml Reply
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Ayebanifesunday Reply
light is a kind of radiation That stimulates sight brightness a source of illumination.
Electromagnet radiation creates space 7th, 8th, and 9th dimensions at the rate of c.
That is the reason that the speed of light is constant.
This creation of new space is "Dark Energy".
The first two sets of three dimensions, 1 through 6, are "Dark Matter".
As matter decays into luminous matter, a proton, a neutron, and an electron creat deuterium.
There are three sets of three protons, 9.
There are three sets of three neutrons, 9.
A free neutron decays into a proton, an electron, and a neutrino.
There are three sets of five neutrinoes, 15.
Neutrinoes are two dimensional.
A positron is composed of the first three dimensions.
An electron is composed of the second three dimensions.
What is photoelectric
Hsssan Reply
light energy (photons) through semiconduction of N-P junction into electrical via excitation of silicon purified and cristalized into wafers with partially contaminated silicon to allow this N-P function to operate.
i.e. Solar pannel.
Photoelectric emission is the emission of electrons on a metal surface due to incident rays reflected on it
If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
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it has everything to do with the angle the UV sunlight strikes your sunglasses.
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
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1.79×10_¹⁹ km per hour
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কাজের একক কী
কাজের একক কী
Energy is ability so capacity to do work.
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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