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

 Page 4 / 12

The probability density of finding a classical particle between x and $x+\text{Δ}x$ depends on how much time $\text{Δ}t$ the particle spends in this region. Assuming that its speed u is constant, this time is $\text{Δ}t=\text{Δ}x\text{/}u,$ 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 $\text{Δ}x$ 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 $n={\left(K\text{/}{E}_{1}\right)}^{1\text{/}2}$ .

## Solution

The kinetic energy of the cart is

$K=\frac{1}{2}m{u}^{2}=\frac{1}{2}\left(0.40\phantom{\rule{0.2em}{0ex}}\text{kg}\right){\left(0.50\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)}^{2}=0.050\phantom{\rule{0.2em}{0ex}}\text{J}.$

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

${E}_{1}=\frac{{\pi }^{2}{\hslash }^{2}}{2m{L}^{2}}=\frac{{\pi }^{2}{\left(1.05\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-34}\text{J}·\text{s}\right)}^{2}}{2\left(0.40\phantom{\rule{0.2em}{0ex}}\text{kg}\right){\left(2.0\phantom{\rule{0.2em}{0ex}}\text{m}\right)}^{2}}=1.700\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-68}\text{J}.$

Therefore, $n={\left(K\text{/}{E}_{1}\right)}^{1\text{/}2}={\left(0.050\text{/}1.700\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-68}\right)}^{1\text{/}2}=1.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{33}$ .

## 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.

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 $x=L\text{/}4$ ? (b) Repeat question (a) for a classical particle.

a. 9.1%; b. 25%

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

${\text{Ψ}}_{n}\left(x,t\right)={e}^{\text{−}i{\omega }_{n}t}{\psi }_{n}\left(x\right)=\sqrt{\frac{2}{L}}{e}^{\text{−}i{E}_{n}t\text{/}\hslash }\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\frac{n\pi x}{L},n=1,2,3,...$

A 10kg mass lift to a height of 24m and release. what is the total energy of the system
mechanics is that branch of physical and mathatics that
what is the difference between a molecule and atom
Atoms are single neutral particles. Molecules are neutral particles made of two or more atoms bonded together.
Manfred
what I'd dynamic propulsion
A body quadruples its momentum when its speed doubles.What was the initial speed in units of c.i.e..what was u/c ?
what is enthalpy?
a thermodynamic quantity equivalent to the total heat content of a system
RAMLA
proparty of tharmo dainamic
bloch
What is the meaning of Nuclear Fission?
what do you mean by dynamics single particles
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
what is light
light is a kind of radiation That stimulates sight brightness a source of illumination.
kenneth
Electromagnet radiation creates space 7th, 8th, and 9th dimensions at the rate of c.
John
That is the reason that the speed of light is constant.
John
This creation of new space is "Dark Energy".
John
The first two sets of three dimensions, 1 through 6, are "Dark Matter".
John
As matter decays into luminous matter, a proton, a neutron, and an electron creat deuterium.
John
There are three sets of three protons, 9.
John
There are three sets of three neutrons, 9.
John
A free neutron decays into a proton, an electron, and a neutrino.
John
There are three sets of five neutrinoes, 15.
John
Neutrinoes are two dimensional.
John
A positron is composed of the first three dimensions.
John
An electron is composed of the second three dimensions.
John
What is photoelectric
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.
Michael
i.e. Solar pannel.
Michael
Photoelectric emission is the emission of electrons on a metal surface due to incident rays reflected on it
Benita
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?
it has everything to do with the angle the UV sunlight strikes your sunglasses.
Jallal
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.
Jallal
I hope I'm clear if I'm not please tell me to clarify further or rephrase
Jallal
what is bohrs model for hydrogen atom
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
1.79×10_¹⁹ km per hour
Swagatika
3×10^8
Benita
what r dwarf planet
what is energy
কাজের একক কী
Jasim
কাজের একক কী
Jasim
Energy is ability so capacity to do work.
kenneth