Abhijit Das

Numerade Educator
Researcher

Biography

Hi All, Myself Abhijit doing my PhD in Center for High Energy Physics, Indian Institute of Science. I have a passion in Physics teaching and love explore life in many ways.

Education

Abhijit has not yet added their education credentials.

Educator Statistics

Numerade tutor for 5 years
21 Students Helped

Topics Covered

Mastering Motion: Achieving Efficiency Along a Straight Line
Mastering Newton's Laws: Tips for Applying Them Effectively
Exploring the Fascinating World of Quantum Physics

Abhijit's Textbook Answer Videos

06:58
Introduction to Quantum Mechanics

(a) Show that the set of all square-integrable functions is a vector space (refer to Section A.1 for the definition). Hint: The main problem is to show that the sum of two square-integrable functions is itself square-integrable. Use Equation 3.7. Is the set of all normalized functions a vector space?
(b) Show that the integral in Equation $3.6$ satisfies the conditions for an inner product (Section A.2).

Chapter 3: Formalism
Section 1: Hilbert Space
Abhijit Das
05:33
Introduction to Quantum Mechanics

(a) For what range of $v$ is the function $f(x)=x^{\prime \prime}$ in Hilbert space, on the interval $(0.1)$ ? Assume $v$ is real, but not necessarily positive.
(b) For the specific case $v=1 / 2$, is $f(x)$ in this Hilbert space? What about $x f(x) ?$ How about $(d / d x) f(x)$ ?

Chapter 3: Formalism
Section 1: Hilbert Space
Abhijit Das
07:59
Introduction to Quantum Mechanics

Show that if $\langle h \mid \hat{Q} h\rangle=\langle\hat{Q} h \mid h\rangle$ for all functions $h$ (in Hilbert space), then $\langle f \mid \hat{Q} g\rangle=\langle\hat{Q} f \mid g\rangle$ for all $f$ and $g$ (i.e., the two definitions of "hermitian" - Equations $3.16$ and $3.17-$ are equivalent). Hint: First let $h=f+g$, and then let $h=f+i g$.

Chapter 3: Formalism
Section 2: Observables
Abhijit Das
12:52
Introduction to Quantum Mechanics

(a) Show that the sum of two hermitian operators is hermitian.
(b) Suppose $\hat{Q}$ is hermitian, and $\alpha$ is a complex number. Under what condition (on $\alpha$ ) is $\alpha \hat{Q}$ hermitian?
(c) When is the product of two hermitian operators hermitian?
(d) Show that the position operator $(\hat{x}=x)$ and the hamiltonian operator $(\hat{H}=$ $\left.-\left(\hbar^{2} / 2 m\right) d^{2} / d x^{2}+V(x)\right)$ are hermitian.

Chapter 3: Formalism
Section 2: Observables
Abhijit Das
09:19
Introduction to Quantum Mechanics

The hermitian conjugate (or adjoint) of an operator $\hat{Q}$ is the operator $\hat{Q}^{\dagger}$ such that
$$
\langle f \mid \hat{Q} g\rangle=\left\langle\hat{Q}^{\dagger} f \mid g\right\rangle \quad(\text { for all } f \text { and } g)
$$
(A hermitian operator, then, is equal to its hermitian conjugate: $\hat{Q}=\hat{Q}^{\dagger} .$ )
(a) Find the hermitian conjugates of $x, i$, and $d / d x$.
(b) Construct the hermitian conjugate of the harmonic oscillator raising operator. $a_{+}$ (Equation 2.47).
(c) Show that $(\hat{Q} \hat{R})^{\dagger}=\hat{R}^{\dagger} \hat{Q}^{\dagger}$.

Chapter 3: Formalism
Section 2: Observables
Abhijit Das
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