QM: system with two lin. independent states

In summary, the conversation involved a discussion of a system with two linear independent states and a 2x2 Hamiltonian matrix. The time-independent Schroding equation was used to find the eigenvalues and eigenvectors of the Hamiltonian. One member was stuck on solving the equation, but eventually solved it with the help of a friend.
  • #1
jacobrhcp
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[SOLVED] QM: system with two lin. independent states

Homework Statement



Imagine a system with just two linear independent states:

[tex]|1>=(1,0)[/tex] and [tex]|2>=(0,1)[/tex] (these are actually column matrices, but I don't know how to type those in tex)

[tex]|\Psi>=a|1>+b|2>=(a,b)[/tex], also [tex]|a|^2+|b|^2=1[/tex]

suppose the hamiltonian is a 2x2 matrix with entries j,g above and g,j below, [tex](g,j \in R>0)[/tex].

The time-independent schroding equation reads

[tex]H |\Psi>=i h/(2 pi) d/dt(|\Psi>)[/tex]

a) find the eigenvalues and eigenvectors of this hamiltonian
b) suppose the system starts out at t=0 in |1>, what is the state at time t?

The Attempt at a Solution



I thought of solving the time-independent SE;

[tex]ja+gb= i h/(2 pi) d/dt(a)[/tex]
[tex]ga+jb= i h/(2 pi) d/dt(b)[/tex]

but does this mean g=0? And if so, how do I get any further. I'm stuck.
 
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  • #2
Change to the basis where H is diagonal.
 
  • #3
solved it with a friend today =)
 

What is a "QM: system with two lin. independent states"?

A "QM: system with two lin. independent states" refers to a quantum mechanical system that has two linearly independent states, meaning that the states are not correlated or dependent on each other. In quantum mechanics, states are the possible configurations or properties of a system, and linear independence indicates that these states can exist separately and do not affect each other.

What are independent states in quantum mechanics?

In quantum mechanics, independent states are states that are not correlated or dependent on each other. This means that the states can exist separately and do not affect each other. In a quantum mechanical system, the states are the possible configurations or properties of the system, and independence between states indicates that the system can exist in multiple states simultaneously.

How does a "QM: system with two lin. independent states" behave?

A "QM: system with two lin. independent states" behaves according to the principles of quantum mechanics. This means that the system can exist in multiple states simultaneously, and the states can change or evolve over time according to the laws of quantum mechanics. The system can also exhibit properties such as entanglement and superposition.

What is the significance of a "QM: system with two lin. independent states"?

A "QM: system with two lin. independent states" has significance in understanding the behavior and properties of quantum mechanical systems. It allows scientists to study the principles of quantum mechanics, such as superposition and entanglement, in a controlled and measurable way. These systems also have practical applications in technologies such as quantum computing and cryptography.

How are "QM: system with two lin. independent states" studied and observed?

"QM: system with two lin. independent states" can be studied and observed using various experimental techniques in quantum mechanics, such as quantum state tomography and quantum interferometry. These techniques involve manipulating and measuring the states of the system to understand its behavior and properties. The results of these experiments can provide valuable insights into the fundamental principles of quantum mechanics.

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