Chapter 5: Q19P (page 229)

Find the energy at the bottom of the first allowed band, for the case $β = 10$ , correct to three significant digits. For the sake of argument, assume $\frac{α}{a} = 1$ eV.

Short Answer

Expert verified

The energy at the bottom of the first band is 0.345eV.

Step by step solution

Define the Schrödinger equation

A differential equation that describes matter in quantum mechanics in terms of the wave-like properties of particles in a field. Its answer is related to a particle's probability density in space and time.
The time-dependent Schrodinger equation is represented as

$I ħ \frac{d}{d t} | ψ (t) > = \hat{H} | ψ (t) >$

Calculating the minimum energy of first band

The minimum energy of the first band required z,f(z)=1

And f(z) is given by

$f (z) = \cos z + β \frac{\sin z}{z} z = ka, z \leq πβ = 10 = \frac{mαa}{h^{2}} \Rightarrow f (z) = \cos z + 10 \frac{\sin z}{z}$

Using MATLAB to calculate the minimum energy of first band

$Using MATLAB we get z = 2.6276 . Energy is given with E = \frac{h^{2} k^{2}}{2 m} = \frac{h^{2}}{2 m} {(\frac{z}{a})}^{2} = \frac{z^{2}}{2 a} \frac{h^{2}}{mβ} \frac{α}{α} = \frac{z^{2}}{2 a} \frac{α}{β} \frac{α}{a} = 1 eV E = \frac{2 . 62767^{2}}{20} 1 eV E = 0.345 eV$

Therefore the energy at the bottom of the first band is 0.345eV.

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Find the energy at the bottom of the first allowed band, for the case $β = 10$ , correct to three significant digits. For the sake of argument, assume $\frac{α}{a} = 1$ eV.

Short Answer

Step by step solution

Define the Schrödinger equation

Calculating the minimum energy of first band

Using MATLAB to calculate the minimum energy of first band

One App. One Place for Learning.

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Find the energy at the bottom of the first allowed band, for the caseβ=10 , correct to three significant digits. For the sake of argument, assume αa=1eV.

Short Answer

Step by step solution

Define the Schrödinger equation

Calculating the minimum energy of first band

Using MATLAB to calculate the minimum energy of first band

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Radiation

Fields in Physics

Magnetism and Electromagnetic Induction

Engineering Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Find the energy at the bottom of the first allowed band, for the case $β = 10$ , correct to three significant digits. For the sake of argument, assume $\frac{α}{a} = 1$ eV.