{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Fund Quantum Mechanics Lect &amp; HW Solutions 31

# Fund Quantum Mechanics Lect &amp; HW Solutions 31 - 2.6...

This preview shows page 1. Sign up to view the full content.

2.6. HERMITIAN OPERATORS 13 Answer: By definition of Inv, and then using [1, p. 43]: Inv sin( kx ) = sin( kx ) = sin( kx ) Inv cos( kx ) = cos( kx ) = cos( kx ) So by definition, both are eigenfunctions, and with eigenvalues 1 and 1, respectively. 2.6 Hermitian Operators 2.6.1 Solution herm-a Question: A matrix A is defined to convert any vector vectorr = x ˆ ı + y ˆ into vectorr 2 = 2 x ˆ ı +4 y ˆ . Verify that ˆ ı and ˆ are orthonormal eigenvectors of this matrix, with eigenvalues 2, respectively 4. Answer: Take x = 1 ,y = 0 to get that vectorr = ˆ ı transforms into vectorr 2 = 2ˆ ı . Therefore ˆ ı is an eigenvector, and the eigenvalue is 2. The same way, take x = 0 ,y = 1 to get that ˆ transforms into 4ˆ , so ˆ is an eigenvector with eigenvalue 4. The vectors ˆ ı and ˆ are also orthogonal and of length 1, so they are orthonormal. In linear algebra, you would write the relationship vectorr 2 = Avectorr out as: parenleftBigg x 2 y 2 parenrightBigg = parenleftBigg 2 0 0 4 parenrightBiggparenleftBigg x y parenrightBigg = parenleftBigg 2 x 4 y parenrightBigg In short, vectors are represented by columns of numbers and matrices by square tables of
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}