Problem 7 If you look at the wavefunction that describes a particle in a quantum system,...
Problem 7
If you look at the wavefunction that describes a particle in a
quantum system, list two things that
you can conclude/deduce from the wavefunction.
Homework Answers
from any wavefunction of a particle in a quantum state we can calculate the it belong to which orbital , number of nodes in the wave fuction and average or expectation value for any operater like energy , position , momentum . we can also calculate the probability density by this .
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Problem 7
If you look at the wavefunction that describes a particle in a
quantum system,...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
Problem 1. Consider a system of three identical particles. Each particle has 5 quantum states with...
Problem 1. Consider a system of three identical particles. Each particle has 5 quantum states with energies 0, ε, 2E, 3E, 4E. For distinguishable particles, calculate the number of quantum states where (1) three particles are in the same single-particle state, (2) only two particles are in the same single-particle state, and (3) no two particles are in the same single-particle state. Problem 2. For fermions, (1) calculate the total number of quantum states, and (2) the number of states...
QUESTION 1: In quantum mechanics, the behaviour of a quantum particle (like an electron, for example)...
QUESTION 1: In quantum mechanics, the behaviour of a quantum particle (like an electron, for example) is described by the Schrödinger equation. The time-independent Schrödinger equation can be written in operator notation as H{y(x, y, z))-Ey(x, y, z) where H is known as the Hamiltonian operator and is defined as h2 2m Here, is a positive physical) constant known as Planck's constant and m is the mass of the particle (also Just a constant). V(x,y,Z) is a real-valued function. The...
Question category: quantum mechanics, quantum chemistry 10. (1 mark) Look at the following normalized (real) wavefur...
Question category: quantum mechanics, quantum
chemistry
10. (1 mark) Look at the following normalized (real) wavefur ook at the following normalized (real) wavefunctions, plotted as "(x) versus x, for a particle in a box -1 < x < 1. 1.25 0.751 0.5 0.25 -0.25 -0.5 005 -0.5 0 0 1 -0.5 0.5 Which wavefunction (A, B or C) will yield the largest value for <x>? Justify your answer using only ONE grammatically correct sentence. (Hint: you should be able to...
Please help with this question it’s from quantum, thank you so much! A particle of mass...
Please help with this question it’s from quantum, thank you so
much!
A particle of mass m in an infinite potential well of length a starts out with a wavefunction where 4,1 and ψ2 are the first two stationary states with energies Ei and E. respectively, and and Describe the motion of the probability density. Be specifict For example, if it oscilates, what is the angular frequency of the oscillation in terms of the mass m and the box length...
09 Estimate the ground state energy and wavefunction for a particle in a box using the...
09 Estimate the ground state energy and wavefunction for a particle in a box using the variational method with the following trial wavefunction, where N is the normalization constant and ß is a variational parameter that should be minimized. 14) = N exp(-Bx2) (7.6) 1. Is this a good trial wavefunction for this approximation (justify your answer)? 2. Why is this not a good wavefunction? 3. Can you solve this problem both analytically and numerically? Pay careful attention to limits...
7. Compare the functional form of the wavefunction derived for a particle confined to a 2D...
7. Compare the functional form of the wavefunction derived for a particle confined to a 2D plane, Y., , to one confined to a 1D line, Y., or Y... Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related. Model 2 The following are the normalized solutions to the Schrödinger equations for a particle confined to a plane defined in Model 1. V., ,(x,y) = Pnevy »* Vaba sin2= sin2,...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, w...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested...
Consider a quantum particle in a 3D box that is not a cube. It has side...
Consider a quantum particle in a 3D box that is not a cube. It has side lengths: a = 1 Å b = 1 Å c = 2 Å Answer the following: 1. Derive the wave vector k in the terms of nx, ny, and nz 2. find the equation for energy as a function of nx, ny, and nz 3. List the five lowest energies a particle can have in this system and list all the different states for...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
Problem 1. Consider a system of three identical particles. Each particle has 5 quantum states with energies 0, ε, 2E, 3E, 4E. For distinguishable particles, calculate the number of quantum states where (1) three particles are in the same single-particle state, (2) only two particles are in the same single-particle state, and (3) no two particles are in the same single-particle state. Problem 2. For fermions, (1) calculate the total number of quantum states, and (2) the number of states...
QUESTION 1: In quantum mechanics, the behaviour of a quantum particle (like an electron, for example) is described by the Schrödinger equation. The time-independent Schrödinger equation can be written in operator notation as H{y(x, y, z))-Ey(x, y, z) where H is known as the Hamiltonian operator and is defined as h2 2m Here, is a positive physical) constant known as Planck's constant and m is the mass of the particle (also Just a constant). V(x,y,Z) is a real-valued function. The...
Question category: quantum mechanics, quantum
chemistry
10. (1 mark) Look at the following normalized (real) wavefur ook at the following normalized (real) wavefunctions, plotted as "(x) versus x, for a particle in a box -1 < x < 1. 1.25 0.751 0.5 0.25 -0.25 -0.5 005 -0.5 0 0 1 -0.5 0.5 Which wavefunction (A, B or C) will yield the largest value for <x>? Justify your answer using only ONE grammatically correct sentence. (Hint: you should be able to...
Please help with this question it’s from quantum, thank you so
much!
A particle of mass m in an infinite potential well of length a starts out with a wavefunction where 4,1 and ψ2 are the first two stationary states with energies Ei and E. respectively, and and Describe the motion of the probability density. Be specifict For example, if it oscilates, what is the angular frequency of the oscillation in terms of the mass m and the box length...
09 Estimate the ground state energy and wavefunction for a particle in a box using the variational method with the following trial wavefunction, where N is the normalization constant and ß is a variational parameter that should be minimized. 14) = N exp(-Bx2) (7.6) 1. Is this a good trial wavefunction for this approximation (justify your answer)? 2. Why is this not a good wavefunction? 3. Can you solve this problem both analytically and numerically? Pay careful attention to limits...
7. Compare the functional form of the wavefunction derived for a particle confined to a 2D plane, Y., , to one confined to a 1D line, Y., or Y... Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related. Model 2 The following are the normalized solutions to the Schrödinger equations for a particle confined to a plane defined in Model 1. V., ,(x,y) = Pnevy »* Vaba sin2= sin2,...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested...
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