A quantum particle is trapped in a quantum system, giving it three discrete set of energy states. These states are represented as vectors in R³ and their directions are unchanged under the matrix -14 20 2 –4 10 4 -4 4 -8 that computes their corresponding energies. Specifically, if v is a state of the particle, then its energy is given by the scalar for which its length is changed under an application of H. (Note that a knowledge of physics is not required to solve this problem. You only need to deal with vectors and matrices using the information given in the questions.) H (a) Find the possible energies of H. [Hint: One of them is 6.] (b) Find a diagonal matrix A and an invertible matrix P so that HP = PA. (c) Find a diagonal matrix D and an invertible matrix Q so that H³ + H = QDQ-¹.
A quantum particle is trapped in a quantum system, giving it three discrete set of energy states. These states are represented as vectors in R³ and their directions are unchanged under the matrix -14 20 2 –4 10 4 -4 4 -8 that computes their corresponding energies. Specifically, if v is a state of the particle, then its energy is given by the scalar for which its length is changed under an application of H. (Note that a knowledge of physics is not required to solve this problem. You only need to deal with vectors and matrices using the information given in the questions.) H (a) Find the possible energies of H. [Hint: One of them is 6.] (b) Find a diagonal matrix A and an invertible matrix P so that HP = PA. (c) Find a diagonal matrix D and an invertible matrix Q so that H³ + H = QDQ-¹.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![A quantum particle is trapped in a quantum system, giving it three discrete set of energy states.
These states are represented as vectors in R³ and their directions are unchanged under the
matrix
H =
-14 20 2
-4 10
4
-4
4-8
that computes their corresponding energies. Specifically, if v is a state of the particle, then its
energy is given by the scalar for which its length is changed under an application of H. (Note
that a knowledge of physics is not required to solve this problem. You only need to deal with
vectors and matrices using the information given in the questions.)
(a) Find the possible energies of H. [Hint: One of them is 6.]
(b) Find a diagonal matrix A and an invertible matrix P so that HP = PA.
(c) Find a diagonal matrix D and an invertible matrix Q so that H³ + H = QDQ-¹.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d44c7ff-ac66-483d-829c-187b6e606263%2F875d46f9-1ede-49dc-bdfd-5d0e14c39066%2Fi8j27d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A quantum particle is trapped in a quantum system, giving it three discrete set of energy states.
These states are represented as vectors in R³ and their directions are unchanged under the
matrix
H =
-14 20 2
-4 10
4
-4
4-8
that computes their corresponding energies. Specifically, if v is a state of the particle, then its
energy is given by the scalar for which its length is changed under an application of H. (Note
that a knowledge of physics is not required to solve this problem. You only need to deal with
vectors and matrices using the information given in the questions.)
(a) Find the possible energies of H. [Hint: One of them is 6.]
(b) Find a diagonal matrix A and an invertible matrix P so that HP = PA.
(c) Find a diagonal matrix D and an invertible matrix Q so that H³ + H = QDQ-¹.
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