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2 2 m d 2 d x 2 ( B k sin ( k x ) ) = E ( B k sin ( k x ) ) .

Computing these derivatives leads to

E = E k = 2 k 2 2 m .

According to de Broglie, p = k , so this expression implies that the total energy is equal to the kinetic energy, consistent with our assumption that the “particle moves freely.” Combining the results of [link] and [link] gives

E n = n 2 π 2 2 2 m L 2 , n = 1 , 2 , 3 , . . .

Strange! A particle bound to a one-dimensional box can only have certain discrete (quantized) values of energy. Further, the particle cannot have a zero kinetic energy—it is impossible for a particle bound to a box to be “at rest.”

To evaluate the allowed wave functions that correspond to these energies, we must find the normalization constant B n . We impose the normalization condition [link] on the wave function

ψ n ( x ) = B n sin n π x / L
1 = 0 L d x | ψ n ( x ) | 2 = 0 L d x B n 2 sin 2 n π L x = B n 2 0 L d x sin 2 n π L x = B n 2 L 2 B n = 2 L .

Hence, the wave functions that correspond to the energy values given in [link] are

ψ n ( x ) = 2 L sin n π x L , n = 1 , 2 , 3 , . . .

For the lowest energy state or ground state energy    , we have

E 1 = π 2 2 2 m L 2 , ψ 1 ( x ) = 2 L sin ( π x L ) .

All other energy states can be expressed as

E n = n 2 E 1 , ψ n ( x ) = 2 L sin ( n π x L ) .

The index n is called the energy quantum number    or principal quantum number    . The state for n = 2 is the first excited state, the state for n = 3 is the second excited state, and so on. The first three quantum states (for n = 1 , 2 , and 3 ) of a particle in a box are shown in [link] .

The wave functions in [link] are sometimes referred to as the “states of definite energy.” Particles in these states are said to occupy energy levels    , which are represented by the horizontal lines in [link] . Energy levels are analogous to rungs of a ladder that the particle can “climb” as it gains or loses energy.

The wave functions in [link] are also called stationary state     s and standing wave state     s . These functions are “stationary,” because their probability density functions, | Ψ ( x , t ) | 2 , do not vary in time, and “standing waves” because their real and imaginary parts oscillate up and down like a standing wave—like a rope waving between two children on a playground. Stationary states are states of definite energy [ [link] ], but linear combinations of these states, such as ψ ( x ) = a ψ 1 + b ψ 2 (also solutions to Schrӧdinger’s equation) are states of mixed energy.

The first three quantum states of a quantum particle in a box for principal quantum numbers n=1, n=2, and n=3 are shown: Figure (a) shown the graphs of the standing wave solutions. The vertical axis is the wave function, with a separate origin for each state that is aligned with the energy scale of figure (b). The horizontal axis is x from just below 0 to just past L. Figure (b) shows the energy of each of the states on the vertical E sub n axis. All of the wave functions are zero for x less than 0 and x greater than L. The n=1 function is the first half wave of the wavelength 2 L sine function and its energy is pi squared times h squared divided by the quantity 2 m L squared. The n=2 function is the first full wave of the wavelength 2 L sine function and its energy is 4 pi squared times h squared divided by the quantity 2 m L squared. The n=3 function is the first one and a half waves of the wavelength 2 L sine function and its energy is 9 pi squared times h squared divided by the quantity 2 m L squared.
The first three quantum states of a quantum particle in a box for principal quantum numbers n = 1 , 2 , and 3 : (a) standing wave solutions and (b) allowed energy states.

Energy quantization is a consequence of the boundary conditions. If the particle is not confined to a box but wanders freely, the allowed energies are continuous. However, in this case, only certain energies ( E 1 , 4 E 1 , 9 E 1 , …) are allowed. The energy difference between adjacent energy levels is given by

Δ E n + 1 , n = E n + 1 E n = ( n + 1 ) 2 E 1 n 2 E 1 = ( 2 n + 1 ) E 1 .

Conservation of energy demands that if the energy of the system changes, the energy difference is carried in some other form of energy. For the special case of a charged particle confined to a small volume (for example, in an atom), energy changes are often carried away by photons. The frequencies of the emitted photons give us information about the energy differences (spacings) of the system and the volume of containment—the size of the “box” [see [link] ].

Questions & Answers

A 10kg mass lift to a height of 24m and release. what is the total energy of the system
ADEPOJU Reply
mechanics is that branch of physical and mathatics that
ADEPOJU
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.
Manfred
what I'd dynamic propulsion
Elias Reply
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
RAMLA
proparty of tharmo dainamic
bloch
What is the meaning of Nuclear Fission?
Benita Reply
what do you mean by dynamics single particles
Peacekamei Reply
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
what is light
Ayebanifesunday Reply
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
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.
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?
Rakhi Reply
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
Swagatika Reply
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
Propessor Reply
1.79×10_¹⁹ km per hour
Swagatika
3×10^8
Benita
what r dwarf planet
Sivalakshmi Reply
what is energy
Isiguzo Reply
কাজের একক কী
Jasim
কাজের একক কী
Jasim
Energy is ability so capacity to do work.
kenneth
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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