Introduction
Suppose we consider the similar particle as in chapter but this time it is constrained to shift in a rectangular box of dimensions a, b and c in length. Within the box (i.e. between x= 0 and a; y = 0 and b and z = 0 and c), the potential energy is zero. At the walls and everywhere outside the box, the potential is.
Definition of Particle in a Three-Dimensional Box
The Schrodinger wave equation for 3 dimensional (3D) box is
δ2φ/δx2 + δ2φ/δy2 + δ2φ/δz2 + 8 π2m/h2(E-V) φ =0
where and V are (x,y,z).
since V = 0 inside the box, then the equation becomes
δ2φ/δx2 + δ2φ/δy2 + δ2φ/δz2 + 8 π2m/h2Eφ=0
Equation may be solved by writing the wave function as the product of 3 functions, each depending on one coordinate.
φ(x, y, z) = X(x)Y(y)Z(z)
differentiating equation gives
δφ/δx = Y (y) Z (z) δx/δx
δ2φ/δx2= Y (y) Z(z) δ2x/δx2
and by a similar reasoning
δ2φ/δy2 = X (x)Z(z) δ2y/δy2
δ2φ/δz2 = X(x)Y(y) δ2z/δz2
Y(y)Z(z)δ2x/δx2 + X(x)Z(z) ) δ2y/δy2 + X(x)Y(y) δ2z/δz2 + 8 π2m/h2 EX(x)Y(y)Z(z) = 0 dividing all through by Y(y)X(x)Z(z) one obtains
-h2/8 π2 m* 1/X(x) δ2x/δx2 +1/Y(y) * δ2y/δy2 + 1/Z(z) δ2z/δz2 = E
We can write the energy as the sum of three contributions associated with the coordinates.
E = Ex + Ey + Ez
using equation, we can divide the expression attained into 3 equations.
-h2/8 π2 m *1/x δ2x/δx2 = Ex
-h2/8 π2 m *1/y δ2y/δy2 = Ey
-h2/8 π2 m *1/z δ2z/δz2 = Ez
Each of the last three equations is similar to the expression for the particle in a one-dimensional box discussed in chapter. Hence their solutions are respectively:
X = [√(2/a)]sin nπx/a
En,x = n2xh2/8ma2
Y = [√(2/b)]sin nπy/b
En,y = n2y h2/ 8mb2
X = [√(2/c)] sin nπx/c
En,z = n2z h2/ 8mc2
Where a, b, c are length in x, y, z directions, correspondingly. Also, nx, ny, nz are the quantum numbers correspondingly.
Since φ (x, y, z) = X(x)Y(y)Z(z) and E = Ex + Ey + Ez, then
ψ (x,y,z) = √a/v sin(nπx/a) sin(nπy/b) sin(nπz/c)
where V is the volume of the box.
Ex,y,z = h2/δm * n2x/a2 + n2y/b2 [+ n2z/c2]
Whenever the three dimensional box has geometrical symmetry, more interesting results are often obtained. For example, in a cubic box, a = b =c, thus equation becomes
E = h2/ φm(n2x + n2y + n2z)
Suppose nx =3, ny = nz =2, then
Ψ(x,y,z) = [/(8/v)] sin3πx/a* sin2πy/a*sin2πz/a]
E = h2/8ma2(32 + 22+ 22) = 17h2/8ma2
Assuming we have another set of values nx =2, ny =3, nz =2, then
Ψ(x,y,z) = [/(8/v )]sin 2πx/a sin3πy/a sin2πz/a
E = h2/8ma2 (22 + 32+ 22) = 17h2/8ma2
Suppose nx =2, ny =2, nz =3, then
Ψ(x,y,z) = [/( 8/v)]sin2πx/a sin2πy/a sin3πz/a
E = h2/8ma2(22 + 22+ 32) = 17h2/8ma2
Even though such states are different, the values of the energies are the same. The three states are said to be degenerate since they have equal energy.
For a situation where nx = ny = nz = 1, it corresponds to only one state of the system. The same is true of nx = ny = nz = 2, but situations such as nx ≠ ny = nz three degenerate states are obtained as shown in the Figure below.
Fig: Quantized Energy Levels of a Particle in a Cubic Box
Suppose we wish to calculate the transition energy between the levels E2,2.2 and E3,2,1 then the energy difference is calculated as
ΔE = hν = 14h2/8ma2 - 12h2/8ma2 = 2h2/8ma2 = h2/4ma2
Given suitable data, it should be possible for us to calculate the frequency (ν) of the transition between the two states.
Zero point energy
According the old quantum theory, the energy level of a harmonic oscillator is E = hv.
The lowest energy level via n = 0 would have zero energy. Depend on the wave treatment of the system, the energy level corresponds to the state with quantum numbers nx = ny = nz =1. The dissimilarity between such 2 values is termed the zero point energy.
Tutorsglobe: A way to secure high grade in your curriculum (Online Tutoring)
Expand your confidence, grow study skills and improve your grades.
Since 2009, Tutorsglobe has proactively helped millions of students to get better grades in school, college or university and score well in competitive tests with live, one-on-one online tutoring.
Using an advanced developed tutoring system providing little or no wait time, the students are connected on-demand with an expert at https://www.tutorsglobe.com. Students work one-on-one, in real-time with a tutor, communicating and studying using a virtual whiteboard technology. Scientific and mathematical notation, symbols, geometric figures, graphing and freehand drawing can be rendered quickly and easily in the advanced whiteboard.
Free to know our price and packages for online chemistry tutoring. Chat with us or submit request at [email protected]
tutorsglobe.com eye assignment help-homework help by online receptor organs tutors
An electronic device which retrieves, stores, and processes data, and can be programmed with instructions is known as computer.
Heredity and Environment tutorial all along with the key concepts of Probability in Genetics, Probability to the Formation of Gametes, Formation of Offspring in Humans, Formation of Offspring in Plants
tutorsglobe.com saving water assignment help-homework help by online fresh water crisis and management tutors
Avail comprehensive support to get rid of assignment problems with Particle Physics Assignment Help tutors.
build up the layout of a lap winding for a 3 phase ac machine comprising 4 pole and 24 slots. there are 2 coil sides per slot.
uses of transistors in stabilized power supplies tutorial all along with the key concepts of Solid state voltage regulators, Operational amplifier voltage regulator, non-inverting input of operational amplifier, Zener voltage, output current of circuit, voltage regulator Integrated Circuits
tutorsglobe.com basic techniques of plant tissue culture assignment help-homework help by online plant tissue culture tutors
A case of historic US (United States) use of the identical word Radiophone presents in the 1960s. This word is employed to explain push-to-talk two-way equipment and systems.
We offer supreme Theoretical Physics Assignment Help service with 100% surety in terms of uniqueness at feasible prices.
We have finest tutors who are committed to offer exceptional Bimolecular Assignment Help service at budget-friendly prices ensuring top grades.
Theory and lecture notes of Market signaling all along with the key concepts of market signaling, Education as a signal, Educational Signaling Game, Economics of information. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Market signaling.
Seeking for consistent Psychopathology Assignment Help service at rational prices? Relax and let PhD tutors do it for you!
In quest for top-class Complex analysis Assignment Help? Hire us and clear all your qualms with apt tutors and secure A++
tutorsglobe.com natural methods of vegetative propagation assignment help-homework help by online reproduction in angiosperms tutors
1943974
Questions Asked
3689
Tutors
1441149
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!