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Fjolvar
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Hello,
I am having some difficulty with my Quantum Mechanics homework so I will post the questions and answer them to the best of my ability, which is still at a beginners level since this subject is new to me. I will update my answers as I figure them out. Any elaboration on the questions or my answers is greatly appreciated. Thank you.
1. Answer the following in brief sentences:
a. The time independent Schrodinger Equation applies to only a particular class of systems. The Hamiltonians of all these systems have a common property. What is this property?
The Hamiltonians do not depend on time.
b. If the Hamiltonian satisfies the equation: H(x) = H(-x), what can we say about the eigenfunctions of the Hamiltonian?
If [tex]\Phi[/tex](x) is a solution, so is [tex]\Phi[/tex](-x).
c. A wave equation is solved and a relationship between the frequency [tex]\omega[/tex] and the wave number k (=2*pi/[tex]\lambda[/tex], with [tex]\lambda[/tex] being the wavelength) is obtained. A wave packet is formed using waves with k values near k0. What is the speed of the wave packet?
I'm not sure how to approach this question.
2. Consider an Infinite one-dimensional potential well:
A particle of mass m is released in the well at time t=0 with its initial state given by the wave function.
a. A measurement of its position is made, what are the possible results for x? For each measured value, what is the probability density?
A measurement for x is possible between its boundaries such as 0<x<a.
I'm not sure how to calculate the probability density.
b. If instead of its position, its energy is measured, What are the possible results for E? For each measured value of E, What is the probability?
I don't know.
c. Does <x (t)> depend on time? Calculate <x (t)>.
No. <x> = a/2
d. Does <E (t)> depend on time? Calculate <E (t)>.
No. E = n^2*pi^2*hbar^2 / 2*m*a^2
e. If the particle is in the ground state and its momentum is measured, what are the possible results of the measurement? What is the associated probability for each measured value?
I don't know.
I am having some difficulty with my Quantum Mechanics homework so I will post the questions and answer them to the best of my ability, which is still at a beginners level since this subject is new to me. I will update my answers as I figure them out. Any elaboration on the questions or my answers is greatly appreciated. Thank you.
1. Answer the following in brief sentences:
a. The time independent Schrodinger Equation applies to only a particular class of systems. The Hamiltonians of all these systems have a common property. What is this property?
The Hamiltonians do not depend on time.
b. If the Hamiltonian satisfies the equation: H(x) = H(-x), what can we say about the eigenfunctions of the Hamiltonian?
If [tex]\Phi[/tex](x) is a solution, so is [tex]\Phi[/tex](-x).
c. A wave equation is solved and a relationship between the frequency [tex]\omega[/tex] and the wave number k (=2*pi/[tex]\lambda[/tex], with [tex]\lambda[/tex] being the wavelength) is obtained. A wave packet is formed using waves with k values near k0. What is the speed of the wave packet?
I'm not sure how to approach this question.
2. Consider an Infinite one-dimensional potential well:
A particle of mass m is released in the well at time t=0 with its initial state given by the wave function.
a. A measurement of its position is made, what are the possible results for x? For each measured value, what is the probability density?
A measurement for x is possible between its boundaries such as 0<x<a.
I'm not sure how to calculate the probability density.
b. If instead of its position, its energy is measured, What are the possible results for E? For each measured value of E, What is the probability?
I don't know.
c. Does <x (t)> depend on time? Calculate <x (t)>.
No. <x> = a/2
d. Does <E (t)> depend on time? Calculate <E (t)>.
No. E = n^2*pi^2*hbar^2 / 2*m*a^2
e. If the particle is in the ground state and its momentum is measured, what are the possible results of the measurement? What is the associated probability for each measured value?
I don't know.
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