Science Answer Note #55

Question 1 Q12-01] We have a circuit with a single qubit created with the code given below. What should replace "#Your code here" if we want to apply a NOT operator to the qubit? q = QuantumRegister(1) c = ClassicalRegister(1) qc = QuantumCircuit(q,c) # Your code here qc.measure(q[0],c[0]) job = AerSimulator().run(qc,shots=1024) counts = job.result().get_counts(qc) print(counts) Answer [Q12-01] To apply a NOT operator to a single qubit circuit, we can use the Pauli-X matrix, which is commonly known as the NOT gate. This can be represented as a rotation around the X-axis by a pi radians, and it flips the state of the qubit from 0 to 1 and vice versa. In the given code, the " Question 2 derive the wave function and energy levels for the electrons in the h2+ species. Answer The H2+ species, also known as dihydrogen cation, is a diatomic ion composed of two protons and one electron. In order to derive the wave function and energy levels for the electrons in this species, we must first understand the properties of a diatomic ion and the concept of wave functions. A diatomic ion is a molecule composed of only two atoms, one of which has gained or lost one or more electrons to become an ion. In the case of H2+, one of the hydrogen molecules has lost an electron to form a positively charged ion. This results in a strong electrostatic attraction between the two positively charged protons and the lone electron. The wave function, denoted by the Greek letter Psi (Ψ), is a mathematical function that describes the location and behavior of an electron in an atom or molecule. It is used to calculate the probability of finding an electron in a particular region of space. To derive the wave function for the electrons in the H2+ species, we will use the BornOppenheimer approximation, which assumes that the nuclei (protons) are much heavier than the electrons and thus move much slower. This allows the Schrödinger equation, a fundamental equation in quantum mechanics, to be separated into nuclear and electronic components. Using the one-dimensional Schrödinger equation, we can express the wave function for H2+ as follows: Ψ(x1,x2) = Ψn1(x1)Ψn2(x2) Where Ψn1 and Ψn2 are the ground state wave functions for the two individual hydrogen nuclei (protons). These wave functions can be calculated using the formula: Ψn(x) = (2/n)^(1/2)sin(nπx/L) Where n is the quantum number, x is the position of the electron in the hydrogen molecule, and L is the interatomic distance between the two protons. Now, to calculate the energy levels for the electrons in H2+, we will use the Rydberg formula: E = -Rhc/n^2 Where R is the Rydberg constant, h is Planck's constant, c is the speed of light, and n is the principle quantum number representing the energy level. Substituting these values into the formula and solving for n, we get the following energy levels for the electron in H2+: n = 1, 2, 3, ... Question 3 Which statements about the Schrödinger equation are true? * The Schrödinger equation gives the energy and position of an electron in an atom. The electron simultaneously behaves as a wave and a particle. Since the electron has wave properties, equations describing three-dimensional waves could be used to derive the Schrödinger equation. Only some solutions to the Schrödinger equation will describe orbitals. Solving a wave function gives quantum numbers that uniquely describe each orbital in an atom. Answer The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of non-relativistic particles, such as electrons, in a given potential. It is named after Austrian physicist Erwin Schrödinger, who first proposed this equation in 1926. There are several statements about the Schrödinger equation that are true and can be supported by academic sources. Firstly, the Schrödinger equation gives the energy and position of an electron in an atom. This is known as the wave function, which describes the probability of finding the electron at a particular position in space. This concept is essential in understanding the electronic structure of atoms. Secondly, the Schrödinger equation describes the electron as simultaneously behaving as a wave and a particle. This is a fundamental principle in quantum mechanics called the wave-particle duality, which explains the physical behavior of particles at a microscopic level. Furthermore, the Schrödinger equation is derived from equations describing threedimensional waves, such as the wave equation and the de Broglie equation. This shows the interconnection between classical and quantum mechanics and how classical concepts can be applied to understand quantum phenomena. It is also true that only some solutions to the Schrödinger equation describe the orbitals of an atom. This is because the Schrödinger equation has an infinite number of solutions, but only specific solutions, called eigenfunctions, are physically meaningful. These eigenfunctions correspond to the orbitals of an atom, and they give the quantum numbers that uniquely describe each orbital. In order to solve the Schrödinger equation and obtain the wave function of an atom, we need to apply a process called normalization. This process ensures that the wave function is a valid description of the electron's behavior and follows the principles of quantum mechanics. The detailed calculations for solving the Schrödinger equation and obtaining the wave function can be found in many chemistry textbooks, such as "Physical Chemistry" by Peter Atkins and Julio de Paula. In conclusion, the Schrödinger equation is a powerful tool in predicting the behavior of particles in a given potential and plays a crucial role in understanding the electronic structure of atoms. Its study requires a deep understanding of both classical and quantum mechanics, and it continues to be a subject of ongoing research and exploration in the field of chemistry. Question 4 What is the energy difference ΔE between the l=0 and l=1 rotational states of diatomic hydrogen? Use 1.05×10−34J⋅ s for ℏ . The equilibrium separation of the two hydrogen atoms in diatomic hydrogen is 0.074nm . The mass of a single hydrogen atom is 1.67×10−27kg . Express the energy difference in joules to two significant figures. Answer According to the Bohr model of the hydrogen atom, the energy levels of a diatomic hydrogen molecule can be calculated using the formula: E= -13.6 eV/n^2, where n is the principal quantum number. In rotational states, the principal quantum number (n) is equal to the rotational quantum number (l), hence, for l=0, n=1 and for l=1, n=2. To calculate the energy difference between these two rotational states, we can use the following formula: ΔE= E(final state) - E(initial state). Therefore, for l=0, E(initial state)= -13.6 eV and for l=1, E(final state)= -3.4 eV. Substituting the values in the formula, we get ΔE= (-3.4 eV) - (-13.6 eV) = 10.2 eV. To convert this energy difference from electron volts (eV) to joules (J), we use the conversion factor: 1 eV = 1.602 x 10^-19 J. Therefore, the energy difference ΔE in joules would be (10.2 eV) x (1.602 x 10^-19 J/eV) = 1.63 x 10^-18 J. However, to express the final answer to two significant figures, we round off the value to 2 significant figures, which gives us ΔE= 1.6 x 10^-18 J. Question 5 If a sample starts with 1.2 nanograms of C14, about how old will it be when it contains only 0.075nanograms of C14? Answer To accurately determine the age of the sample based on the given information, we must understand the concept of carbon dating and how it works. Carbon dating, also known as radiocarbon dating, is a method used to estimate the age of organic materials by measuring the levels of carbon-14 (C14) present in the sample. As carbon-14 decays over time, this technique can be used to calculate the approximate age of the sample. To answer the question, we will use the half-life formula of carbon-14 to calculate the age of the sample. The half-life of carbon-14 is approximately 5730 years, which means that after 5730 years, the amount of carbon-14 in a sample will decrease by half. Step 1: Calculate the half-life of carbon-14 from 1.2 nanograms to 0.075 nanograms 5730 years = 1 half-life (from 1.2 nanograms to 0.6 nanograms) 5730 years = 2 half-lives (from 0.6 nanograms to 0.3 nanograms) 5730 years = 3 half-lives (from 0.3 nanograms to 0.15 nanograms) 5730 years = 4 half-lives (from 0.15 nanograms to 0.075 nanograms) Step 2: Calculate the age of the sample by multiplying the half-life by the number of halflives. 5730 years x 4 half-lives = 22,920 years Therefore, the age of the sample would approximately be 22,920 years when it contains only 0.075 nanograms of carbon-14.