Reference request for operator theory in Quantum mechanics

Question

I am studying Shankar's Principles of Quantum Mechanics. In the first chapter where the author introduces the necessary mathematics tool for QM, the concept of derivatives of operators with respect to parameters is presented. According to the text, let $\theta(\lambda)$ be an operator that depends on $\lambda$, assume that $\theta(\lambda)=e^{\lambda\Omega}$ (where $\Omega$ is Hermitian), then $$\frac{d\theta(\lambda)}{dt}=\Omega e^{\lambda\Omega}=\theta(\lambda)\Omega$$ The problem is that differentiating an operator like a function is all new to me, and I don't know how the theory comes about. The author did not go further about the topic as QM is the main priority. However, I would like to know more about it. Unfortunately, I don't know where to start as I don't know which field of mathematics this theory belongs to. I hope someone could point me to the right direction and suggest some relevant texts. Thank you.

Answer 1

I can't recommend Mary L. Boas' Mathematical Methods in the Physical Sciences enough. For ALL mathematics relevant to undergraduate physics.

You probably want to break the study of Operator theory into separate parts which most texts in QM do not do.

Consider the space of polynomials in x of degree $<N$, denoted $P_N(x)$. They all have the form,

$$P_N(x)=a_0+a_1x+a_2x^2+...+a_{N-1}x^{N-1}$$

You can think of each $x^n$ as a Vector in its own right. The coefficients are effectively coordinates in the Vector Space Spanned by the vectors $x^n$ where the $x^n$ are Basis Vectors.

The derivative of $ax^n$ is $nax^{n-1}$. Let $a=1$, we have the effect of differentiation on a basis vector. Not that it returns a multiple of the basis vector.

Now Differentiation is a Linear Operator and it maps the space of polynomials into itself. From the coordinate representation of the original polynomial alone, its possible to determine the out come of applying the linear operator. In fact, in the space $P_N(x)$, differentiation can be expressed as the product of a constant matrix applied to a vector presenting some polynomial allowing for multiple direct Linear Algebraic analysis to the polynomials space.

Thes basis composed of the various $x^n$ are not the best to use for certain problems in Quantum Mechanics. For many you want to use Legendre Polynomials. They are Orthogonal on the interval $[-1,1]$. You can find the Legendre polynomials without differential equations. You canuse the Graham-Schmidt Orthonormalization process on the interval [-1,1].

Note everythign in caps. They are some of the most important Linear Algebra concepts you use in Quantum Mechanics.

For somewhat more advanced approaches to Orthogonal Polynomials associated with Differential equations, you want to pay close attention to Rodriques's Formulas for Legendre, Laguerre, and Hamilton polynomials. The approach of Rodrigue gives one unified solution approach for some of the most important problems in undergraduate QM.

You'll find all of this and more in Boas.