Nanoscience HW 1

Calculate the number of gallium atoms and arsenide atoms per cubic centimeter in a bulk GaAs crystal. The lattice constant of GaAs is a=0.565 nm.

1) calculate number of each type of atom in the image 1/8 for corners, 1/2 for face, 1/4 for edges 2) convert lattice constant to usable unit (ie cm) 3)divide #of atoms by lattice constant. (do this for each type of atom)

(10 points) Consider a linear 1, 3, 5, 7-Octatetraene molecule CH2CHCHCHCHCHCHCH2, the absorption of this molecule is at 460 nm, what is the length of the box using the model of the particle-in-a-box in quantum mechanics? Compare your result with the one obtained from the bond lengths data and calculate the relative error? (C-C bond length is 1.54 Angstrom and that of the C=C bond is 1.35 Angstrom, mass of electrons is 9.11 10-31 kg and speed of light is 2.998 108 m s-1)

1) draw chemical 2) count pi electrons 3) estimate based on bond lengths and types of bonds 4) solve for l/a to 5) (actual-theoretical)/theoretical

Consider a two-level system with 𝜀1 = 1.75 × 10−22 J and 𝜀2 = 3.50 × 10−21 J. If 𝑔2 = 2 𝑔1 , what value of temperature is required to obtain 𝑛2/𝑛1 = 0.150 ? what value of temperature is required to obtain 𝑛2/𝑛1 = 0.900 ? What conclusions can you draw from these different ratios at different temperatures? (kB= 1.381 × 10−23 J K -1 )

1) rewrite all the values you know 2) write what you are looking for 3) choose correct equation 4) set up for one the equation for each n1/n2 ratio 5) solve both sets of the same equation to solve for T

Which of the following wave functions are eigenfunctions of the operator 𝑑 2 /𝑑𝑥 2 ? If they are eigenfunctions, what is the eigenvalue? (show your derivations)

1) take first derivative 2) take second derivative 3) see if og function is still there multiplied by a constant 4) if og function is there that is the eigen function and the constant is the eigenvalue

(1) Normalize Ψ(x) = x(a-x) defined over the interval 0 ≤ x ≤ a by multiplying Ψ(x) by a constant N. Please determine N; (2) Determine the expectation value of position for Ψ(x) = x(a-x) over the interval 0 ≤ x ≤ a.

Part 1 1) rewrite the Ψ(x)=x(a-x) to integrate it over 0 to a multiplied by a constant "N" set equal to 1 2) simplify as much as possible ( you can pull a constant outside the integral ie "N2") 3) integrate by doing the anti derivative 4) replace x with a 5) simplify 6)solve for N Part 2 1) apply the expectation value formula 2) A hat equals "x" from the interval 3) set up numerator and denominator 4) simplify num and denom, you can pull a out as a constant 5) integrate num and demon by doing the antiderivative 6)replace x with a 7) simplify 8)solve for expectation value

How to estimate the number of surface atoms in a spherical nanostructure using a unit-cell approach? Please show the process of calculation. (assuming an underlying cubic unit cell with a lattice constant a and the radius of the sphere much larger than the lattice constant a) Hint: the outer layer of unit cells is considered the "surface" of the particles.

Step 1: estimate the number of atoms in a nanostructure Vshell=(4/3pir^3)-(4/3pi(r-a)^3) Step 2: estimate # of surface atoms in a nanostructure N*Vcell=Vshell draw picture

Suppose that the wave function for a system can be written as Ψ(x) = √2 4 ∅1 (𝑥) + 1 √2 ∅2 (𝑥) + 2+√2𝑖 4 ∅3 (𝑥) and that ∅1 (𝑥),∅2 (𝑥), ∅3 (𝑥) are normalized eigenfunctions of the operator Ekinetic with eigenvalues E1, 2E1, and 4E1, respectively. a. Verify that Ψ(x) is normalized; b. What are the possible values that you could obtain in measuring the kinetic energy on identically prepared systems? c. What is the probability of measuring each of these eigenvalues? d. What is the average value of Ekinetic that you would obtain from a large number of measurements?

part a 1) square the constants (eigenvalues) in the wavefunction to see if the add up to 1 (remember to use conjugate is number is complex) (just square if real) part b 1) these are the eigenvalues- the constants so E1, E2 and E3 ( the numbers before the ∅) use there relationships E1 2E1 and 4E1 2) part c 1) part d 1)pa

(15 points) The state of an electron confined to the region between x=0 and x=a is described by the wavefunction: Ψ(x) = 1 2 ∅1 (𝑥) + 1 2𝑖 ∅2 (𝑥) − 1 √2 ∅4 (𝑥), where ∅1 (𝑥) = √ 2 𝑎 sin 𝜋𝑥 𝑎 , ∅2 (𝑥) = √ 2 𝑎 sin 2𝜋𝑥 𝑎 , ∅4 (𝑥) = √ 2 𝑎 sin 4𝜋𝑥 𝑎 . (a) When the energy of the electron is measured, what value or values are measured? Note that the Hamiltonian operator in this case has the form of and in this case the potential energy V(x) can be omitted for convenience. (b) What is the expectation value of the particle's energy,〈𝐸〉 ? Explain.

part a 1) the measured value is energy part b 1) use energy formula

(10 points) It is useful to consider the result for the energy eigenvalues for the ID box 𝐸𝑛 = 𝑛 2ℎ 2 8𝑚𝑎2 , n=1,2,3,...... as a function of n, m and a. (a) By what factor do you need to change the box length to decrease the zero point energy by a factor of 35 for a fixed value of m? (b) For what value of n is the energy greater than the zero point energy by a factor of 250 for fixed values of a and m? (c) By what factor would you have to change a to have the zero point energies of a helium atom be equal to the zero point energy of a hydrogen atom in the box? (mass of a helium atom is 4.002 u and mass of a hydrogen atom is 1.008 u; u is the atomic mass unit)

part a 1) zeropoint n=1 2) 1/35=E1=E2 3) energy formula and solve for l part b 2) part c 3)

What is the difference between surface area and specific surface area (SSA)?

specific surface area includes a ratio of surface area, volume and density but surface area is just the collective areas of each face. Two objects with the same SA could have different SSA's or the other way around.