1807 Words8 Pages

Name

University

Course

Instructor

Date

Quantum Physics Exam 3

Problem1. Hydrogen Essentials

PART A: Make a plot of the energy levels of the hydrogen atom. Plot the energy values En in the vertical direction for n = 1, 2,3,4,5. Plot the orbital angular momentum quantum number in the horizontal direction for l = 0,1,2,3,4. For each n, show every allowed value of l. Label every energy level spectroscopically (1s, 2s, 2p, ...). Indicate the m degeneracy of each l level. Show that the total degeneracy of each En is n2.

Solution:

Bohr radius separates the potential energy of two electrons according to the equation: U=e2/4πε0a0 …Equation [1]. This equation represents one atomic unit of energy.

The Schrödinger equation for the Hydrogen*…show more content…*

The levels(n=1,2,3,4) are represented in black.

How the energy levels are lined up. Explanation:

From the sketch in Figure 1below, for level n=1, there is only one value allowed for l. The value is l=0. the potential well that corresponds to l=1(light blue) is above n = 1. For n =2, there are two values allowed for l. The possible values of l are l=0 and l=1because this energy level is above the wells of l=0 and l=1and below l=2 (green) and l=3 (violet). In other words, there is only one possible orbital (1s) for the first energy level (n=1)). The existence of other superior orbitals (1p,1d,etc) is forbidden by the fact that their energy wells (in which they lie) are above the n=1n=1 energy. Figure 1: Sketch where n = 1

PART C: Sketch just the l = 0 effective potential. Add the locations of the n = 1,2,3 energy levels to your sketch. Sketch the radial wave functions for each of these energy levels and label each wave function with the appropriate Rnl designation. Sketch the corresponding probability distributions per unit volume | (r)|2 dV. Sketch the corresponding radial probability distributions 4⇡r2 | (r) |2*…show more content…*

From Equation [10], the normalized time-dependent wave function at a later time, t=to can be expressed as follows:

{|ψ(x,t=to)}=〖3/√23 ∑▒e〗^(-(iEnt)/h) |ψ1 (x)-〖1/√23 ∑▒e〗^(-(iEnt)/h) |ψ2 (x)+ 〖2/√23 ∑▒e〗^(-(iEnt)/h) |ψ3 (x)- 〖3/√23 ∑▒e〗^(-(iEnt)/h) |ψ3 (x) … Equation [12]

PART D: If you measure the energy at time t, what are the possibilities and what are the probabilities?

University

Course

Instructor

Date

Quantum Physics Exam 3

Problem1. Hydrogen Essentials

PART A: Make a plot of the energy levels of the hydrogen atom. Plot the energy values En in the vertical direction for n = 1, 2,3,4,5. Plot the orbital angular momentum quantum number in the horizontal direction for l = 0,1,2,3,4. For each n, show every allowed value of l. Label every energy level spectroscopically (1s, 2s, 2p, ...). Indicate the m degeneracy of each l level. Show that the total degeneracy of each En is n2.

Solution:

Bohr radius separates the potential energy of two electrons according to the equation: U=e2/4πε0a0 …Equation [1]. This equation represents one atomic unit of energy.

The Schrödinger equation for the Hydrogen

The levels(n=1,2,3,4) are represented in black.

How the energy levels are lined up. Explanation:

From the sketch in Figure 1below, for level n=1, there is only one value allowed for l. The value is l=0. the potential well that corresponds to l=1(light blue) is above n = 1. For n =2, there are two values allowed for l. The possible values of l are l=0 and l=1because this energy level is above the wells of l=0 and l=1and below l=2 (green) and l=3 (violet). In other words, there is only one possible orbital (1s) for the first energy level (n=1)). The existence of other superior orbitals (1p,1d,etc) is forbidden by the fact that their energy wells (in which they lie) are above the n=1n=1 energy. Figure 1: Sketch where n = 1

PART C: Sketch just the l = 0 effective potential. Add the locations of the n = 1,2,3 energy levels to your sketch. Sketch the radial wave functions for each of these energy levels and label each wave function with the appropriate Rnl designation. Sketch the corresponding probability distributions per unit volume | (r)|2 dV. Sketch the corresponding radial probability distributions 4⇡r2 | (r) |2

From Equation [10], the normalized time-dependent wave function at a later time, t=to can be expressed as follows:

{|ψ(x,t=to)}=〖3/√23 ∑▒e〗^(-(iEnt)/h) |ψ1 (x)-〖1/√23 ∑▒e〗^(-(iEnt)/h) |ψ2 (x)+ 〖2/√23 ∑▒e〗^(-(iEnt)/h) |ψ3 (x)- 〖3/√23 ∑▒e〗^(-(iEnt)/h) |ψ3 (x) … Equation [12]

PART D: If you measure the energy at time t, what are the possibilities and what are the probabilities?

Related

## Mole Mass Lab Report

372 Words | 2 Pages2. Explain how to determine the formula mass (mole mass) for a compound. Molar mass is the mass (in grams) of one mole of a substance. Using the atomic mass of an element and multiplying it by the conversion factor grams per mole (g/mol), you can calculate the molar mass of that element. First, find the chemical formula for the compound.

## John Dalton Research Paper

808 Words | 4 PagesThose are the orbitals surrounding the atom where the electrons are. Atoms are made of electrons, protons, and neutrons, and the number of those depends on the element. The amount of electrons and protons depends on the element’s atomic number where the atomic number is equal to the number of protons, which is equal to the

## Atomic Theory Research Paper

1675 Words | 7 PagesDalton’s atomic theory contained the following ideas: All atoms of a given element are identical. The atoms of different elements vary in mass and size. Atoms are indestructible. Chemical reactions may result in their rearrangement, but not their creation or destruction. Dalton also outlined a law of multiple proportions, which described how reactants will combine in set ratios.

## Relative Atomic Mass Lab Report

380 Words | 2 PagesExplain what is meant by relative atomic mass?- Relative atomic mass is the measure of the mass of an element which consists of the protons and neutrons; it is always the bigger number and has the symbol Ar. An obvious example of this is oxygen has the Ar of 16 yet sodium has the Ar of 23. Explain how relative formula mass is calculated using the periodic table- You need to find the relative atomic mass (M) for each atom but this can only be done when the relative atomic mass have been found, then you add all of the masses in the compound together to find the RFM. Sodium Hydroxide is the example we used within our experiment. Na= 23 O= 16 H= 1 NaOH 23 + 16 + 1 = 40 NaOH Mr= 40 Explain how it is possible to work out amounts of substance

## Unit 8 Lab: Photoelectric Effect Created By Young Su Kang

430 Words | 2 PagesUnit 8 LAB- Photoelectric Effect Created by Young Su Kang Materials: • Google PhET Simulation- photoelectric effect Vocabulary: • Electron volt– a unit of energy that is equal to the energy of an electron that is accelerated by a potential difference of 1 volt • Frequency- the number of times something happens in a given period of time • Intensity- a measure of how much energy passes through a region in a given time • Photoelectric effect- the emission of electrons from the surface of a material when struck by electromagnetic radiation • Photon- the smallest possible amount of light; a quantum of light. • Voltage- a measure of the strength of an electrical field. • Wavelength- the distance between consecutive crests or troughs of a wave • Work function- the minimum energy required to remove an electron from a solid

## History Of Atom Research Paper

776 Words | 4 Pagestwo different directions to the spin. He also gave the world a rule to dictate how an electron in an atom would behave. Pauli’s exclusion principal states that, “if an electron has a certain set of quantum numbers then no other electron in that atom can have the same set of quantum numbers.” When 1926 came Erwin Schrödinger another Austrian physicist built upon Einstein and Louis de Broglie’s idea of wave particle duality. He developed a mathematical Equation that related the pattern of electrons with that of waves, he gave his wave mechanics the symbol psi . In 1932 James Chadwick discovered a second particle in the nucleus, he fired alpha particles at beryllium and found that neutrons were released.

## Nanofullerene Research Paper

908 Words | 4 PagesThe computationally predicted various possible conformers are shown in Fig.1. The optimized molecular structure with the numbering of atoms of the title compound is shown in Fig.2. The most optimized structural parameters were also calculated by HF/ B3LYP have depicted in Table 1. Quantum chemical calculation was used for NFN to carry out the optimized geometry with the Gaussian 03W program [10] using the B3LYP and HF functional [11, 12] supplemented with standard 3-21G* basis set. Density Functional Theory (DFT) can be used to calculate an accurate electronic structure, HOMO and LUMO energies, Mulliken charge of atoms, energetic orbital levels, global hardness, chemical potential and electrophilicity of systems, and finally chemical, physical properties of fullerene and fullerene derivatives.

## Pros And Cons Of Sulphur

1234 Words | 5 PagesSulphur is a yellowish, non-metallic chemical element in the periodic table. It is found in group 6 and period 3. Sulphur is identified by the letter S. Since Sulphur is the 16th Element of the periodic table therefore atomic number is always similar as the proton number which is located in the nucleus and converts the nucleus in to a positive charge as neutron is a neutral charge (+-) so positive (+) plus neutral (+-) equals a positive charge. The atomic mass for sulphur is 32 amu or (Atomic Mass Unit is the international system of units). The melting point of sulphur is 112.8 °C and the boiling point of Sulphur is 444.6 °C.

## Second Order Reaction Rate Constant Of Ibuprofen With Atrazine Lab Report

1141 Words | 5 Pages1.1 Kinetic model To determine the second order reaction rate constant of Acesulfame K with the different transient species studied, two pairs of independent competition kinetics were established for each transient: Acesulfame K with Ibuprofen and Acesulfame K with Atrazine. Assuming the first pair of competition for the hydroxyl radical generated by NaNO3 irradiation is Acesulfame and Ibuprofen (ACE, IBP). Their respective reaction rates are (M s-1): (Eq. 6) (Eq. 7) With k and k’ the second order reaction rates of Ace and IBP with HO•.

## Periodic Table Research Paper

805 Words | 4 PagesThese metals have only one electron in their outer shell. This means they are ready to lose that one electron in ionic bonding with other elements. As with all metals, the alkali metals are malleable, ductile, and are good conductors of heat and electricity. The alkali metals are softer than most other metals. Cesium and francium are the most reactive elements in this group.

### Mole Mass Lab Report

372 Words | 2 Pages### John Dalton Research Paper

808 Words | 4 Pages### Atomic Theory Research Paper

1675 Words | 7 Pages### Relative Atomic Mass Lab Report

380 Words | 2 Pages### Unit 8 Lab: Photoelectric Effect Created By Young Su Kang

430 Words | 2 Pages### History Of Atom Research Paper

776 Words | 4 Pages### Nanofullerene Research Paper

908 Words | 4 Pages### Pros And Cons Of Sulphur

1234 Words | 5 Pages### Second Order Reaction Rate Constant Of Ibuprofen With Atrazine Lab Report

1141 Words | 5 Pages### Periodic Table Research Paper

805 Words | 4 Pages