Solutions Manual For Quantum Theory of Materials By
Efthimios Kaxiras; John D. Joannopoulos
Time Independent Schrodinger Equation - ANSWER: ĤΨ = EΨ for a particle of mass
M moving in one diversion with energy E
-(ħ^2/2m)(d^2Ψ/dx^2) + V(x)Ψ = EΨ
What is v(x) (TISE) - ANSWER: Potential energy of the particle at point x
What is ħ (TISE) - ANSWER: ħ = h / 2xpi = 1.05457 x 10^-34 js
Ψ wavefunction - ANSWER: Function obtained by solving Schrödinger equation
Contains all dynamical info of system
For each wavefunction solution - ANSWER: That describes motion of the electron
there is a corresponding total energy
Spatial wavefunction - ANSWER: Wave function describing the spatial distribution of
a particle
Time Independent Schrodinger Equation solution - ANSWER: Ψ =e^ ikx where k =
[ 2m(E - V)/ħ^2]^1/2
Ψ =e^ ikx is also written as - ANSWER: Ψ = coskx + isinkx
What is coskx or sinkX in relation to wave function - ANSWER: A wave of wavelength
landa = 2 pi/k
(By comparing coskx with standard harmonic wave cos(2piX/landa)
Kinetic energy of the particle Ek - ANSWER: E-V
Linear momentum and wavelength of the wavefunction relation - ANSWER: P= h/
landa
Principle tenet of quantum mechanics - ANSWER: The wave function contains all the
dynamical info of the system
Born interpretation of the wavefunction - ANSWER: The value of the square modulus
of the wavefunction at a point is proportional to the probability of finding the
particle in a region around the point
If Ψ is real - ANSWER: Then |Ψ|^2 = Ψ^2
If Ψ is complex - ANSWER: Then |Ψ|^2 = Ψ*Ψ
, Ψ* - ANSWER: Complex conjugate (real and always positive)
How to form complex conjugate of complex function - ANSWER: Replace i by -i eg:
e^ikx = e^-ikx
Why is wave function a probability amplitude - ANSWER: Its square modulus is a
probably density
For a 1D system, if the wave function of a particle has the value Ψ at some point x,
then probability of finding the particle between x and x + dx is proportional to -
ANSWER: |Ψ|^2 dx
How to find probability of finding particle in a region - ANSWER: Multiply |Ψ|^2 by
length of region dx
For a particle moving in 3D Ψ depends on - ANSWER: a point R ( x,y, Z)
Ψ(r) - ANSWER: If the wavefunction of a particle has the value Ψ at some location r
then the probability of finding the particle in an infinitesimal volume (dt = dxdydz)
(cube vol) at point r is proportional to the product |Ψ|^2dt at point r
Can use sphere also where dt= 4/3 pi ro^3 (ro=radius)
normalization constant N - ANSWER: If Ψ is a solution of TISE then so is NΨ (Ψ occurs
in every term of TISE )
For NΨ, the probability that a particle is in the region dx is... - ANSWER: (NΨ*)
(NΨ)dx
(N is real)
Sum over all space of the probability of the particle being in the region dx for NΨ is...
- ANSWER: 1 ie: it must be somewhere
How to normalise a wave function + why - ANSWER: Find N (we wont ever do this +
ensures Ψ has factor that puts it in 1D
In 3D Ψ is normalised if - ANSWER: ∫Ψ*Ψ dt =1 where integral is all space accessible
to particle
spherical polar coordinates - ANSWER: (r, θ, Φ) x=rsinθcosΦ; y=rsinθsinΦ; z=rcosθ
Spherical polar coordinate range (r) - ANSWER: R=radius 0 → infinity (from centre
out)
Spherical polar coordinate range (θ) - ANSWER: Colatitude O-pi (from pole to pole)
Spherical polar coordinate range (Φ) - ANSWER: Azimuth O-2pi (around the
circumference)
Efthimios Kaxiras; John D. Joannopoulos
Time Independent Schrodinger Equation - ANSWER: ĤΨ = EΨ for a particle of mass
M moving in one diversion with energy E
-(ħ^2/2m)(d^2Ψ/dx^2) + V(x)Ψ = EΨ
What is v(x) (TISE) - ANSWER: Potential energy of the particle at point x
What is ħ (TISE) - ANSWER: ħ = h / 2xpi = 1.05457 x 10^-34 js
Ψ wavefunction - ANSWER: Function obtained by solving Schrödinger equation
Contains all dynamical info of system
For each wavefunction solution - ANSWER: That describes motion of the electron
there is a corresponding total energy
Spatial wavefunction - ANSWER: Wave function describing the spatial distribution of
a particle
Time Independent Schrodinger Equation solution - ANSWER: Ψ =e^ ikx where k =
[ 2m(E - V)/ħ^2]^1/2
Ψ =e^ ikx is also written as - ANSWER: Ψ = coskx + isinkx
What is coskx or sinkX in relation to wave function - ANSWER: A wave of wavelength
landa = 2 pi/k
(By comparing coskx with standard harmonic wave cos(2piX/landa)
Kinetic energy of the particle Ek - ANSWER: E-V
Linear momentum and wavelength of the wavefunction relation - ANSWER: P= h/
landa
Principle tenet of quantum mechanics - ANSWER: The wave function contains all the
dynamical info of the system
Born interpretation of the wavefunction - ANSWER: The value of the square modulus
of the wavefunction at a point is proportional to the probability of finding the
particle in a region around the point
If Ψ is real - ANSWER: Then |Ψ|^2 = Ψ^2
If Ψ is complex - ANSWER: Then |Ψ|^2 = Ψ*Ψ
, Ψ* - ANSWER: Complex conjugate (real and always positive)
How to form complex conjugate of complex function - ANSWER: Replace i by -i eg:
e^ikx = e^-ikx
Why is wave function a probability amplitude - ANSWER: Its square modulus is a
probably density
For a 1D system, if the wave function of a particle has the value Ψ at some point x,
then probability of finding the particle between x and x + dx is proportional to -
ANSWER: |Ψ|^2 dx
How to find probability of finding particle in a region - ANSWER: Multiply |Ψ|^2 by
length of region dx
For a particle moving in 3D Ψ depends on - ANSWER: a point R ( x,y, Z)
Ψ(r) - ANSWER: If the wavefunction of a particle has the value Ψ at some location r
then the probability of finding the particle in an infinitesimal volume (dt = dxdydz)
(cube vol) at point r is proportional to the product |Ψ|^2dt at point r
Can use sphere also where dt= 4/3 pi ro^3 (ro=radius)
normalization constant N - ANSWER: If Ψ is a solution of TISE then so is NΨ (Ψ occurs
in every term of TISE )
For NΨ, the probability that a particle is in the region dx is... - ANSWER: (NΨ*)
(NΨ)dx
(N is real)
Sum over all space of the probability of the particle being in the region dx for NΨ is...
- ANSWER: 1 ie: it must be somewhere
How to normalise a wave function + why - ANSWER: Find N (we wont ever do this +
ensures Ψ has factor that puts it in 1D
In 3D Ψ is normalised if - ANSWER: ∫Ψ*Ψ dt =1 where integral is all space accessible
to particle
spherical polar coordinates - ANSWER: (r, θ, Φ) x=rsinθcosΦ; y=rsinθsinΦ; z=rcosθ
Spherical polar coordinate range (r) - ANSWER: R=radius 0 → infinity (from centre
out)
Spherical polar coordinate range (θ) - ANSWER: Colatitude O-pi (from pole to pole)
Spherical polar coordinate range (Φ) - ANSWER: Azimuth O-2pi (around the
circumference)
