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I was recently studying Quantum mechanics from R.Shankar's Principles of Quantum Mechanics. I recently encountered improper vectors, and function and infinte-dimensional vectors. But I got confused at a point:
The book introduced a hermitian operator that takes the derivate of a ket $|{\psi} \rangle$ from the Hermitian operator K which is defined as: $$K_{xx'}=-i\delta'(x-x')$$ Then it solved for the eigenfunction of K:$$\psi_k(x)=Ae^{ikx}$$ where k is the eigenvalue and $\psi_k$ is the corresponding eigenfunction. Then the book proved: $$\langle k|k'\rangle=\delta(k-k')$$ here k depicts eigenfunction in vector form. The book then Introduces the X operator and does some maths(I didn't understood a thing,except that it multiplies a function by x.). Then the book says:
In the k basis, K operator just multiplies with the function with k while the X operator becomes $i\frac{d}{dx}$.
I wanted to give it a try,$$K_{kk'}=\langle k|K|k'\rangle$$ $$K_{kk'}=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{\langle k|x\rangle\langle x|K|x'\rangle\langle x'|k'\rangle dxdx'}}$$ $$K_{kk'}=C\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{\delta(x-x')e^{-ikx}e^{ik'x'}dx'dx}}$$ $$K_{kk'}=C\int_{-\infty}^{\infty}{e^{-ikx}e^{ik'x}dx}$$ which corresponds to:$$K_{kk'}=\langle p|p'\rangle$$ Which is not accurate(The above integral was taken from the book only).The Calculation for X in k basis was even more terrible. So here is my question:

  • What is the X operator actually?
  • What is wrong with my calculations above? I hope you guys will help me, I want to know Where I am getting it wrong, I hope you ideas will help. Thanks in advance.
Quantum Mechanic
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Charu _Bamble
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2 Answers2

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The integral in question is exactly a definition of the Dirac delta function $$\delta(z-z^\prime)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i q(z-z^\prime)}dq.$$ I have specifically used different variables $(z, z^\prime, q)$ than the ones in the question because this is true for any variables.

When taking integrals from the book, make sure that the variables match. On one side you have $k$ and $k^\prime$ while on the other there is $p$ and $p^\prime$ - either they are the same, or you must use the definitions for how they are related to each other (and the factor of $2\pi$ might work out nicely).

Quantum Mechanic
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The last integral is equal to a Dirac delta function of k prime minus k. In position space the X operator is the multiply by x operator, while in k space it is the pure imaginary differentiate by k operator. Similarly, in k space the K operator is the multiply by k operator and in position space it is the purely imaginary differentiate by p operator. As a consequence of Fourrier mathematics, the eigenfunctions of the momentum in position space have the same form as the eigenfunctions of position in momentum space.

Albertus Magnus
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