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A detailed explanation of how to solve the Schrödinger equation for a particle in a one-dimensional box. It covers the general procedure for solving the time-independent Schrödinger equation (TISSE), the particle in a 1-D box potential energy, finding the eigenfunctions and eigenvalues, and normalization. The document also includes the calculation of expectation values and properties of the particle in a box.
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Up to this point we have covered the basic ”rules” of quantum mechancis and in the following lectures we will basically repeatedly ”practicing” applications of these rules and approximated methods. It takes some work to familize/internalized quantum mechanics rules. We will start with simple/exactly solvable models. Note that we have gone over some mathematics without doing much practices/examples, particularly, we spent quite some time discuss about ”measurement”, and now we are ready to apply the formalism to real problems. After we have spent some time to practice some simple quantum mechanics problems, we will returm to the issue of fundamental rules of quantum mechanics and explicitly state the ”postulates” or ”ground rules” of quantum mechanic.
Now, lets focus on the applications of quantum mechanics to ”solving” simple problems. We spent some time talking about the important of ”eigenfunction” and ”eigenvalues” of quantum mechanical operators, indeed, we state that a single mea- Eigenfunctions and eigenvalues are impor- tant.
surement causes a wavefunction to collapse into a eigenfunction of the associated observable and results an outcome of a value associated with the eigenfunction. It can’t be emphasized enough that eigenfunctions of an quantum mechanical operator plays the central role in quantum mechanics. Actually, the Schr¨odinger equation is an eigenvalue problem.
Hψˆ = Eψ, Hˆ = pˆ
2 2 m + Vˆ (x). (5.1)
⇒ eigenfunctions of Hˆ, i.e. energy eigenstates, are important ⇒ they represent Hˆ also governs time evolution.
Some declaration or no- tations: Aφ^ ˆ 1 = a 1 φ 1 , this is a eigenproblem, a 1 is a eigenvalue and φ 1 is a eigenfunction. Eigen- function = wavefunc- tion = state.
the likely stable/stationary ”energy” states. (In other words, eigenstate of Hˆ are energetically stable states)
The simplest quantum mechanical system, hence always the first problem to solve in a quantum mechanic class, is the particle in a one dimensional box. Consider a particle of mass m constrainted to move in a 1-D box of length a.
Figure 5.1: Superposition of several wave.
The potential energy is:
V (x) =
∞ if x < 0 0 if 0 ≤ x ≤ a ∞ if x > a
En = ℏ
(^2) k 2 2 m =^
2 m ×^
n^2 π^2 a^2 =^
n^2 h^2 8 ma^2.^ (5.12) corresponding eigenfunction:
ψn(x) = B sin(kx) = B sin
( (^) nπx a
To determine B, we consider the normalization condition
1 =
∫ (^) a 0
ψ∗ n(x)ψn(x)dx = |B|^2 ·
∫ (^) a 0
sin^2 ( nπxa )dx (5.14)
∫ (^) a 0
2 cos
( (^2) nπx a
dx
= |B|^2 ·
2 a^ −^
a 2 nπ sin
( (^2) nπx a
∣∣a 0
=^12 |B|^2 a.
∴ B =
a ,^ (5.15) Any arbitrary φ(x) can be written as φ(x) = ∑ n Cnψn(x).
ψn(x) =
a sin^
nπx a.^ (5.16) It is easy to show that
ψn(x)ψm(x)dx = 0 for n 6 = m, that is {ψn} are orthornor- mal. From these eigenfunctions, we can calculate any properties of the particle.
Let’s summerize the systematic way of solving the Schr¨odinger equation :
Let’s plot these eigenfunctions → stationary states.
Figure 5.2: Plot of the wavefunction of particle in a box.
Observation:
This can be genrerlized to higher dimensions, consider a box of dimension a, b, c along x, y, z axis. Thus the Schr¨odinger equation can be written as
− ℏ
2 2 m (^
∂x^2 +^
∂y^2 +^
∂z^2 )ψ(x, y, z) =^ Eψ(x, y, z).^ (5.27)
Note that ψ is a function of three variables, and the variables in the Hamiltonian are independent. i.e. no terms such as (^) ∂x∂ ∂y∂ or xy · · ·. We call this kind of system a ”seperable” system. In general, the solution can be written as a ”product” of indepedent functions Independent degree of freedom → product!! To write in a product is a huge reduction of com- plexity.
ψ(x, y, z) = X(x)Y (y)Z(z). (5.28)
Plugging into the equation, we obtain:
− ℏ
2 2 m [Y^ (y)Z(z)^
∂x^2 X(x) +^ X(x)Z(z)^
∂y^2 Y^ (y) +^ X(x)Y^ (y)^
∂z^2 Z(z)] = EX(x)Y (y)Z(z).
Divide by X(x)Y (y)Z(z) on both sides, we obtain
− ℏ
2 2 m [^
X(x)
∂x^2 X(x) +^
Y (y)
∂y^2 Y^ (y) +^
Z(z)
∂z^2 Z(z)] =^ E^ (5.30) Why it can be expanded in this manner? ⇒
− 2 ℏm^2 ∂x∂^22 X(x) = ExX(x) − 2 ℏm^2 ∂y∂^22 Y (y) = EyY (y) − 2 ℏm^2 ∂z∂^22 Z(z) = Ez Z(z)
, Ex + Ey + Ez = E. (5.31)
The original equation is now separated into three independent equations, and we know the solutions ⇒ each one is a 1-D particle in a box.
The solution for 1-D particle in a box is:
X(x) =
a sin
( (^) nxπx a
Therefore, the wavefunctions of 3-D particle in a box is:
ψnx,ny ,nz (x, y, z) =^ A product state.
abc sin
( (^) nxπx a
sin
( (^) nyπy b
sin
( (^) nz πz c
energy is:
Enx,ny ,nz = h
2 8 m (^
n^2 x a^2 +^
n^2 y b^2 +^
n^2 z c^2 ).^ (5.34) Now the eigenstate are determined by three quantum numbers, i.e. we must assign quantum number for each coordinate to define a state. Observe from the result above, we could find that:
energy → sum, wave function → products Only for independent subsystems! For a cubic box, energy levels on all three directions are equal, so Degenerate: different wavefunctions (eigenfunctions) with the same energy (eigenvalue).
nx ny nz
The first column in the right hand side is the ground state. The second to forth column represent first excited states and it is three-fold degenerate.