Solve Confusing Matrix Operator Equation in Quantum Mechanics

[Beer-monster]

Homework Statement

I'm new to matrix mechanics in quantum mechanics and I admit that linear algebra is the weakest part of my maths toolkit but I have an operator to convert to matrix form. I'm about 85% sure I've done it right, however I can't solve for the eigenvectors. I'm sure I'm missing something obvious and was wondering if someone could help me find it.

The operator is a Hamiltonian of the form.

$$H=c \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right]$$

If I solve for the eigenvalues I get $\lambda = \pm c$

Which seems right. However, if I try to solves for the eigenvectors using $H'=(H-\lambda I)$ and $H'x = 0$ for the eigenvalue +c, I get a set of equations that don't make sense to me.

$$H'=c \left[ \begin{array}{cc} 1-1 & 1 \\ 1 & -1-1 \end{array} \right]= \left[ \begin{array}{cc} 0 & 1 \\ 1 & -2 \end{array} \right]$$

Using x=(x,y), and if I'm reading this right, this would give a set of equations where y=0 but also y=x. This would imply the answer is a null vector which makes no sense.

Can anyone see where I've gone wrong/what I'm missing?

 

[Beer-monster]

Nevermind , apparently I had forgotten how to take a determinant.

Thanks anyway.

 

[Beer-monster]

Stuck again because now I need to solve this equations:

$$(1-\sqrt{2})x+y = 0 ; x+(-1-\sqrt{2})y = 0$$I've been pounding my head against it buit can't seem to find the way, standard Gaussian techniques seem to be leading to a mess.

 

[Phys O]

Just set x or y to one, then you get the value of the other, (1, something) or (something, 1)

 

[paranormal]

I understand that learning new concepts can be challenging and it's completely normal to struggle with certain topics. It's great that you have identified linear algebra as a weak point and are actively seeking help to improve your understanding.

For the specific problem you mentioned, it seems like you have correctly calculated the eigenvalues of the Hamiltonian operator. However, in order to find the corresponding eigenvectors, you need to solve the equation H'x = 0 for each eigenvalue separately.

In this case, for the eigenvalue +c, you correctly calculated H' as [0 1; 1 -2] and the equations you got are x=0 and x=y. This means that the eigenvector for this eigenvalue is [0, y] where y can be any non-zero value.

I would suggest reviewing the concept of eigenvectors and how to solve for them, and also practicing more with matrix operations. It might also be helpful to seek assistance from a tutor or a colleague who is more familiar with linear algebra and quantum mechanics. Keep practicing and don't get discouraged, as understanding these concepts takes time and effort. Good luck!