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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

fundamental note of a vibrating string
fasoyin Reply
what are matter waves? Give some examples
mallam Reply
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
Mathematical expression of principle of relativity
Nasir Reply
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
obia Reply
What is the importance of de-broglie's wavelength?
Mukulika Reply
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
how those weight effect a stable motion at equilibrium
Nonso Reply
how do I differentiate this equation- A sinwt with respect to t
Evans Reply
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry
de broglie wave equation
LoNE Reply
vy beautiful equation
chandrasekhar
what is electro statics
fitsum Reply
when you consider systems consisting of fixed charges
Sherly
Diagram of the derive rotational analog equation of v= u+at
Nnamnso Reply
what is carat
Arnulfo Reply
a unit of weight for precious stones and pearls, now equivalent to 200 milligrams.
LoNE
a science that deals with the composition, structure, and properties of substances and with the transformations that they undergo.
LoNE
what is chemistry
Mrs Reply
what chemistry ?
Abakar
where are the mcq
Fred Reply
ok
Giorgi
acids and bases
Navya
How does unpolarized light have electric vector randomly oriented in all directions.
Tanishq Reply
unpolarized light refers to a wave collection which has an equal distribution of electric field orientations for all directions
pro
In a grating, the angle of diffraction for second order maximum is 30°.When light of wavelength 5*10^-10cm is used. Calculate the number of lines per cm of the grating.
Micheal Reply
OK I can solve that for you using Bragg's equation 2dsin0over lander
ossy
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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