posted 1 years ago

Explain for the energy of the particle in box with changes in particle mass and box width.

What causes the particle to have discrete (quantized energies)? Hint: The free particle Schrodinger equation is the same as the particle in a box wavefunction, but its energies are not quantized.

You have non-interacting (ideal gas) molecules of mass m in a vessel of width L. You compress the vessel adiabatically (in QM, this means, you compress slowly so that the particles stay in their same quantum states as you compress). Explain how much work was done to compress the gas. Did the temperature change?

posted 1 years ago

b) Write a superposition wavefunction for the particle in a box. Why do we write superposition wavefunctions and not just eigenstate wavefunctions?

\text { Show that ground state wavefunction of the } \mathrm{H} \text { atom, } \psi=N \exp \left(-r / 2 a_{0}\right) \text {, is orthogonal to the } 2 \mathrm{~s} \text { orbital }

\psi=N\left(2-r / a_{0}\right) \exp \left(-r / 2 a_{0}\right) \text { and the } 2 \mathrm{p}_{\mathrm{x}} \text { orbital } \psi=N x \cdot \exp \left(-r / 2 a_{0}\right)

Explain why orthogonality is important in creating superposition wavefunctions.

posted 1 years ago

b) Briefly describe how the general shape of the wavefunction can be described without actually solving the Schrodinger equation.

c) Briefly explain the fundamental relationship between operators and physical observables in quantum mechanics. What is a physical observable and what kinds of operators are associated with physical observables?

d) Describe the composition of a localized particle written as a superposition of momentum states(eigenfunctions of momentum) and how this composition changes as the particle becomes more localized.

posted 1 years ago

Cancelling out equal terms in opposite faces

\frac{\partial \sigma_{x}}{\partial x} d x d y d z+\frac{\partial \tau_{y x}}{\partial y} d x d y d z+\frac{\partial \tau_{z x}}{\partial z} d x d y d z+X d x d y d z=0

So we get the x- equilibrium equation

\frac{\partial \sigma_{x}}{\partial x}+\frac{\partial \tau_{y x}}{\partial y}+\frac{\partial \tau_{z x}}{\partial z}+X=0

\frac{\partial \tau_{y x}}{\partial x}+\frac{\partial \sigma_{y}}{\partial y}+\frac{\partial \tau_{y z}}{\partial z}+Y=0

\frac{\partial \tau_{z x}}{\partial x}+\frac{\partial \tau_{z y}}{\partial y}+\frac{\partial \sigma_{z}}{\partial z}+Z=0

in short, the equilibrium equations in tensor notation

Ou, +X, = 0 (i, j = x, y, z)

Take moment quilibrium about an axis through the center and parallel to z - axis

\tau_{x y} d y d z \frac{d x}{2}+\left(\tau_{x y}+\frac{\partial \tau_{x y}}{\partial x} d x\right) d y d z \frac{d x}{2}-\tau_{y x} d x d z \frac{d y}{2}-\left(\tau_{y x}+\frac{\partial \tau_{y x}}{\partial y} d y\right) d x d z \frac{d y}{2}=0

\tau_{x y} d x d y d z-\tau_{y x} d x d y d z=0

\therefore \tau_{x y}=\tau_{y x}

\tau_{y z}=\tau_{z y}

\tau_{z x}=\tau_{x z}

posted 1 years ago

posted 1 years ago

Wavelength-

Frequency-

Amplitude-

posted 1 years ago

posted 1 years ago

posted 1 years ago

posted 1 years ago