General expectation value
I have a basic question related to finding expectation values of an operator $\hat{Q}$We know that the expectation of $\hat{Q}$ (in the position space) is given by$$\langle Q \rangle=\int {\Psi^* Q\Psi...
View ArticleAnswer by Courage for Doubt in proof of...
No, it is because of differentiationConsider, for example$$f=\frac{1}{2}\sum_{i=0}^n\sum_{j=0}^nq_iq_j=\frac{1}{2}\bigg(\sum_{i=0}^nq_i\bigg)^2$$$$\frac{\partial f}{\partial q_j}=\sum_{i=0}^nq_i$$You...
View ArticleAnswer by Courage for Why is the d'Alembert's Principle formulated in terms...
Consider an system in equilibrium (we can consider a dynamic system too, equilibrium systems are a little simpler). The sum of forces on the system is zero, So the virtual work done is also zero.But...
View ArticleAnswer by Courage for Commutation relation for general case $[a^n, \bar{a}^m]=?$
You can use these relations$$\begin{align}[a,bc] &=[a,b]c+b[a,c]\\ [ab,c] &=a[b,c]+[a,c]b\\ [a,\bar{a}^m] &=[a,\bar{a}]\bar{a}^{m-1}+\bar{a}\left[a, \bar{a}^{m-1}\right]\end{align}$$This is...
View ArticleAnswer by Courage for Quantum Mechanics: how exactly does "delta function...
The delta function is not really a function, it is a distribution,In the strict sense both $\delta (x)$ and $e^{ikx}$ are not normalizable when $n=m$One way to prove your equations is to use fourier...
View ArticleAnswer by Courage for Relation between probability density and transmission...
Yes, the Probability current is defined as $$J(x,t)=\frac{i\hbar}{2m}\bigg(\Psi\frac{\partial \Psi^*}{\partial x}-\frac{\partial \Psi}{\partial x}\Psi^*\bigg)$$If the scattering matrix is defined...
View ArticleAnswer by Courage for Multiplying Lagrangian by a constant
First of all, the Lagrangian is not unique, Multiplying by a constant will give the right Equations of motion (EOM's) when there are no constraint forces.In case of when there are constraints...
View ArticleAnswer by Courage for Show that a linear operator can be written in terms of...
Hint: if $|e_i\rangle$ is the basis any vector $$|\alpha \rangle= \sum a_i |e_i \rangle$$So, $\langle e_i|\alpha \rangle = a_i$ (Fourier inverse)Apply $Q$ on both sides$$Q|\alpha \rangle=\sum_i a_i...
View ArticleAnswer by Courage for How to tell when exponentials are real valued? (Barrier...
In region (1) since $V=0$ the SE becomes$$\frac{h^2}{2m}\frac{\partial^2 \psi}{\partial x^2}=-E\psi$$since $E>0$ we always need $k$ to be real so we take$k=\frac{\sqrt{2mE}}{\hbar}$This will give...
View ArticleAnswer by Courage for Minimum uncertainity
I think I found out the answer to my own question$|f\rangle$ and $|g\rangle$ are defined in Hilbert space which is an inner product space, and one of the properties of this space is $$\langle f|f...
View ArticleMinimum uncertainity
I'm confused in finding the condition for minimum uncertainty, The author in the book I refer goes on saying that$|g\rangle=c|f\rangle$is the condition for minimum uncertainityfor some constant...
View ArticleAnswer by Courage for Why is the Pythagorean Theorem used for error calculation?
The formula$$\frac{\Delta{A}}{A} \approx \frac{\Delta{X}}{X} + \frac{\Delta{Y}}{Y} $$is an approximation because you are ignoring $\Delta X$$\Delta Y$A better approximation would be $$\Delta...
View ArticleAnswer by Courage for Application of Mean Value Theorem to Law of Universal...
Considering the way how $v^2=v_0 ^2-2g\Delta y$ is derived, It is derived through energy conservation, that is$$\frac{1}{2}mv^2-\frac{GMm}{r+\Delta y}=\frac{1}{2}mv_0^2-\frac{GMm}{r}...
View ArticleAnswer by Courage for How to design a simple experiment about force and...
You can show it using a spring based toy car, The toy car has a spring inside, while pulling it backwards you apply a force which winds up the spring, when released the car is propelled forward by the...
View ArticleAnswer by Courage for Why time slows only for moving object and not...
As the person B takes off in a rocket, both of them would see the other clock move at a slower rate, assuming the rocket to be moving at a constant speed, both are in an inertial frames of reference,...
View ArticleAnswer by Courage for Rotational kinematics
The position vector of the particle is $\vec{r}=r\hat{r}$The unit vector along the radius vector $\hat{r}=\{\cos \theta, \sin \theta\} \tag{1}$the unit vector along $\theta$ is...
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Answer by Courage for Center of mass of heavy spring
I had posted an answer earlier but I'm not convinced if it is right, luckily the Link I provided itself had an answer, which is summarized belowLet the spring of Young's modulus $E$, density $\rho$,...
View ArticleAnswer by Courage for Why is the rod moving to and from?
If you carefully observe the video you will see that in the first 2 cases the rod has been placed not exactly at the center, but a little off, this will, as anybody can guess puts uneven weights on the...
View ArticleTo derive the relation between work function and potential energy
I'm reading "The variational principles of mechanics- Lanczos",The author mentions a relation between Work-Function $U(q_1,q_2,\cdots,q_n,\dot q_1,\dot q_2,\cdots,\dot q_n)$ and the potential energy...
View ArticleAnswer by Courage for Are the generalized coordinates in Lagrangian mechanics...
How exactly do the independence of $δy_i$ imply that the coefficients vanish?let $S= \sum_i \bigg(\frac{\partial f}{\partial y_i} - \frac{d}{dx} \frac{\partial f}{\partial \dot{y}_i}\bigg)\ \delta y_i...
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