Question

In: Advanced Math

The following kets name vectors in the Euclidean plane: |a>, |b>, |c>. Some inner products: <a|a>...

The following kets name vectors in the Euclidean plane:

|a>, |b>, |c>.

Some inner products: <a|a> = 1, <a|b> = −1, <a|c> = 0, <b|c> = 1, <c|c> = 1

(a) Which of the kets are normalized?

(b) Which of these are an orthonormal basis?

(c) Write the other ket as a superposition of the two basis kets. What is the norm |h·|·i| of this ket (i.e., the length of the vector)? What is the angle between this ket and the two basis kets?

(d) In the same basis, write as a superposition a ket that has the same direction but is normalized.

Expert Solution

, that means is normalized. imply and are mutually orthogonal. imply is normalized. Let , where are scalers. Now,. But , so . Similarly . So are linarly independent, so forms basis of euclidean plane. Also, they are orthogonal to each other and has norm 1. So it is an orthonormal basis. belong to this plane. , then . So . Then , similarly . So, . . Then . is not normalized. So,

(a) are normalised.

(b) is the orthonormal basis

(c) . . , , . Angle between and is . , , . Angle between and is .

(d) Take, ,

Colby Messinger answered 1 year ago

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