# Search results

1. ### I Momentum eigenfunctions in an infinite well

For an example in a harmonic oscillator energy eigenstate, position x and momentum p both have Gaussian probability distributions around zero. We have tiny but not zero probality to observe any large p as well as x.
2. ### I Momentum eigenfunctions in an infinite well

Each mometum eigenfunction has eigenvalue which is a definite value of momentum of the state.
3. ### B Solving for the Nth divergence in any coordinate system

Why don't you calculate \Delta^2 F by hand and compare it with your formula of n=2 for confirmation ? In notation \partial_j F = F_{,j} \Delta...
4. ### I Momentum eigenfunctions in an infinite well

My complement A classical perfect reflection analogue of \alpha|p'>+\beta|-p'> where p'=\sqrt{2mE'} isn't a solution neither.
5. ### B Solving for the Nth divergence in any coordinate system

Just a quick notice that ##\nabla## here should be Laplacian ##\Delta##.
6. ### I Momentum eigenfunctions in an infinite well

For this exercise for the biginners I prefer observing what happens in finite well potential V system in the limit of ##V \rightarrow \infty## to discussing mathematical subject of ##V = \infty##. We may keep the interval ##[-\infty,+\infty]## and apply usual Fourier transform method to get...
7. ### I Momentum eigenfunctions in an infinite well

In free space system of zero potential energy everywhere, yes.
8. ### I Momentum eigenfunctions in an infinite well

Yes in the well but it is zero outside. That matters. It is superposition of various mometum eigenstates, the coefficiets of which provide probability to observe specific momentum value by product with complex conjugate. Yes inside but it includes no zero potential energy outside. That matters.
9. ### Why do atoms want to complete their shells?

I think covalent bond is the key word for you to understand the issue.
10. ### Trouble solving for end state of two control volumes in a rigid tank

As a first try I take it easy as follows. Say B is vacuum, gas in A expands free adiabatically. Temperature does not change from ##T_A## . ##P_A## drops to ##\frac{V_A}{V_A+V_B}P_A##. Say A is vacuum, gas in B expands free adiabatically. Temperature does not change from ##T_B##. ##P_B##...

12. ### B Solving for the Nth divergence in any coordinate system

Explicitly \Delta^n F=\frac{1}{\sqrt{\vert g\vert}}\partial_{i_1}(\sqrt{\vert g\vert} g^{i_1k_1}\partial_{k_1} \frac{1}{\sqrt{\vert g\vert}}\partial_{i_2}(\sqrt{\vert g\vert} g^{i_2k_2}\partial_{k_2} ... \frac{1}{\sqrt{\vert g\vert}}\partial_{i_n}(\sqrt{\vert g\vert} g^{i_nk_n}\partial_{k_n} F...
13. ### B Solving for the Nth divergence in any coordinate system

If \Delta G=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_kG\right), for any G, then choose G as G= \Delta^{n-1} F I cannot find meaning of odd power of nabla applying scalar function. For example can you show where we meet...
14. ### B Solving for the Nth divergence in any coordinate system

If you have already proved the formula in OP of \Delta F=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_kF\right) I think you have a good reason of recurrence formula \Delta^n F=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert}...
15. ### B Solving for the Nth divergence in any coordinate system

\nabla f=[\frac{\partial }{\partial x}\mathbf{i}+\frac{\partial }{\partial y}\mathbf{j}+\frac{\partial }{\partial z}\mathbf{k}]f , a vector. Its innerproduct with itself is \nabla \cdot \nabla f= [\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}+\frac{\partial^2 }{\partial...
16. ### I Prove that a triangle with lattice points cannot be equilateral

As for 2D lattice we can make one of the lattice points is (0,0) without losing generality. Say other points are ##(n_1,n_2),(m_1,m_2)## n_1^2+n_2^2=A m_1^2+m_2^2=A (n_1-m_1)^2+(n_2-m_2)^2=A where A is square of the side length. You will find contradiction in this set of formla.
17. ### Find the GCD of the given complex numbers

Rather than "the nearest integer", I prefer floor function \lfloor x \rfloor , maximum integer that does not exceed x. \lfloor 1.2 \rfloor=1,\ 1.2=1+0.2 \lfloor -1.4 \rfloor=-2,\ -1.4=-2+0.6 z=(1-2i)+(0.2+0.6i)
18. ### Identifying variables from Riemann sum limits

The last term is 2. For the other sums you shall use the formula 1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}
19. ### Gas temperature in a constant volume

1 Say a wall in the container moves outward for volume v. The gas expands adiabatically \frac{P_2}{P_1}=\frac{V^\gamma}{(V+v)^\gamma}<1 where \gamma = \frac{C_p}{C_v}>1 Then we insert a new partion where there was the wall. The number of gas molecules in the container is N_2=N_1\frac{V}{V+v}...
20. ### Gas temperature in a constant volume

From the equation of state for ideal gas T_1=\frac{P_1 V}{N_1 k} T_2=\frac{P_2 V}{N_2 k} We may get the answer if we know changes of not only P but N.
21. ### Find the Area of the shaded region in the given problem

From your sketch, say S is area we want, it seems 2S=\frac{1}{2}*20^2 (4\alpha)-4*\frac{1}{2}*15\cos \alpha*5-2*\frac{1}{2}*5^2(\pi+2\alpha) where \sin\alpha =\frac{1}{3} Is it same as your idea ?
22. ### Residue Theorem applied to a keyhole contour

Show us residues you calculated, please.
23. ### Rocket acceleration problem: confused about Newton's 2nd Law

I corrected sign in the post recalling R>0. What you point out is mentioned as F(t).
24. ### Rocket acceleration problem: confused about Newton's 2nd Law

I would make 0 = m(t)v(t) + \int_{0}^{t} -\dot m(u) ( v(u)+U(u))~du with U(u)<0 or 0 = m(t)v(t) + \int_{0}^{t} -\dot m(u) ( v(u)-U(u))~du with U(u) >0 for normal thrust. Thus \frac{d}{dt} [m(t)v(t)]=\dot m(t) v(t)+F(t) RHS first term : Minus. Loss of momentum which fuel to be burnt has hold in...
25. ### Rocket acceleration problem: confused about Newton's 2nd Law

Yes, it is in the integral. My typo not dt but du, dummy variable for integration. I corrected it in the post. Thanks.
26. ### Rocket acceleration problem: confused about Newton's 2nd Law

More comprehensive treatment.　Say rocket is at rest at t=0. Momentum of the system is conserved zero. 0=m(t)v(t) + R \int_0^t ( v(u)+U )du differentiating by t, \frac{d}{dt}[m(t)v(t)]+R [ v(t)+U ]=\frac{d}{dt}[m(t)v(t)]+Rv(t)-F=0 where m(t) is mass of rocket body and fuel in tank...
27. ### Rocket acceleration problem: confused about Newton's 2nd Law

m(t) is defined by man made convention of removing ejected fuel mass so d/dt [m(t)v(t)] ##\neq## F does not surprise me. For simplicity say F=0, fuel are just disposed with no thrust or kept in tank but we take off account for m(t). From my post #20 \frac{d}{dt}[m(t)v(t)]+Rv(t)=0 We are...
28. ### Rocket acceleration problem: confused about Newton's 2nd Law

The system consists of Rocket body and fuel. All the fuel is contained in rocket at time t. Some fuel are ejected from Rocket during t and t+##\triangle## t. All the fuel ejected before time t, which continues inertial motion of constant mometum any way, does not matter here. If we choose...
29. ### Rocket acceleration problem: confused about Newton's 2nd Law

No, I meant P is just a momentum value that the system t and the system t+##\triangle## t share by momentum conservation law.
30. ### Rocket acceleration problem: confused about Newton's 2nd Law

At time t P=m(t)v(t) At time t + ##\triangle t## P=m(t+\triangle t) v(t+\triangle t) + R\triangle t \{v(t)+U\} where U is speed of ejected fuel relative to rocket. U < 0, m(t)=m(0)-Rt RHS first term is momentum rocket body and remaing fuel hold. RHS second term is momentum that ejected fuel...

If we omit ##-B^2 y## in RHS, we can solve the simplified ODE, y=B\int_0^t du \ e^{\frac{iA}{\omega}\cos \omega u} We may be able to expect that in a short time the solution of the original ODE does not so much different from it. I would appreciate it If you could check the difference with...
32. ### I Limit of quantum mechanics as h -> 0

Classical Poisson bracket { } , https://en.wikipedia.org/wiki/Poisson_bracket, corresponds with quantum commutator [ ] with \frac{[\ \ ]}{i\hbar} \rightarrow \{\ \ \} in classical limit. ##\frac{\partial V}{\partial x}## comes from classical Poisson bracket.
33. ### B Can't remember the details of an experiment involving small and large envelopes

Hi, not an issue of quantum mechanics but a Bayesian statistics, I know the two envelopes problem, https://en.wikipedia.org/wiki/Two_envelopes_problem
34. ### Integral with different variables

Why don' you try integration by ##\theta## at first. The order should not matter for this exercise at least.
35. ### Integral with different variables

Change of variable ##t=\cos\theta## would make \int_{-1}^1 \frac{dt}{\sqrt{A-Bt}} Does it make sense ?
36. ### Help is needed for converting units of a simple formula

[m3/m3] has no physical dimension but [molecules/cm3] has physical dimension of L^-3. I am afraid some more details of the exercise are required to help you.
37. ### Geometry: prove that point M is touched by 4 circles

Supplement to my post #2 Say three points ##(a_1,a_2),(b_1,b_2),(c_1,c_2)## is on the circle which is caratterized by (l,m, n) \begin{pmatrix} a_1& a_2 & 1 \\ b_1& b_2 & 1 \\ c_1& c_2 & 1 \\ \end{pmatrix} \begin{pmatrix} l \\ m \\ n \\ \end{pmatrix} =- \begin{pmatrix} a_1^2+a_2^2 \\...
38. ### Geometry: prove that point M is touched by 4 circles

Contd. from my previous post The equation of line which connects two commom points of the cirlces i and j, is (l_i-l_j)x+(m_i-m_j)y+n_i-n_j=0 The system of such six lines meet at a point (x,y) are recuced to three equations. \begin{pmatrix} l_1-l_2 & m_1-m_2 & n_1-n_2 \\ l_2-l_3 & m_2-m_3 &...
39. ### B Basic Angle Explanation for Statics

I interpeted which is which as I draw on your sketch below. Does it make sense in the story of the textbook ? What is the title of the textbook ?
40. ### Geometry: prove that point M is touched by 4 circles

Interesting exercise. I take Cartesian coordinates to look at as attached figure. We can write the equation of circle touching three points as x^2+y^2+lx+my+n=0 Giving three (x,y)s of touching points, we get a set (l,m,n) . We get four (l,m,n) sets for the four circles. The system of four...
41. ### Expectation of Kinetic Energy for Deuteron

I am afraid that the potential of deutron is a finite well potential as for radius r as you have used in your calculation rather than 1/r.
42. ### Expectation of Kinetic Energy for Deuteron

So if your coefficent to the formula in post #26 is ##-4\pi A^2 V_0##, we get the same result. I agree with you that the question has something wrong.
43. ### Expectation of Kinetic Energy for Deuteron

I tried =-2\pi V_0 A^2 \int_0^R (1- \cos 2k_1r)r^2dr =-\frac{\pi A^2}{4k_1^3} V_0 \int_0^{2k_1R} (1- \cos y)y^2dy =-\frac{\pi A^2}{4k_1^3} V_0 \{ \frac{u^3}{3}-(u^2-2)\sin u -2u \cos u \} where u=2k_1R Though I am careless in calculation, do you share it with me ? Does it have anything good to...
44. ### A Converting this vector into polar form

I do not go into the mathematics but I am convinced that velocity of fluid at particle surface has no radial component. In fact \mathbf{\hat{r}} \cdot \textbf{v}_s(\hat{\textbf{r}}) = \sum {2\over{{n(n+1)}}} B_n (\mathbf{\hat{r}}\cdot (\hat{\textbf{e}}\cdot \hat{\textbf{r}}) \hat{\textbf{r}} -...
45. ### Find the roots of the complex number ##(-1+i)^\frac {1}{3}##

You are right but not an error because e^{2n\pi i}=1
46. ### Student sliding bag along floor in elevator

The answer seems to make a_x=-\alpha_k (g+a)<0 negative. You can make it positve one with your setting of ##-0.5a_xt^2##.
47. ### Find the roots of the complex number ##(-1+i)^\frac {1}{3}##

Polar form of complex number, -1+i= \sqrt{2}\ e^{3/4\ \pi\ i}, is easy to handle power 1/3. Dividing phase by 3 and multiplying cubic roots of 1, (-1+i)^{1/3}= \sqrt[6]{2}\ e^{1/4\ \pi i + 2/3\ n\pi i} where n=-1,0,1.
48. ### Expectation of Kinetic Energy for Deuteron

Do you see continuing from post #4, ...=-\frac{\hbar^{2}}{2m} \int_{0}^{\infty} dr \psi^* \frac{\partial}{\partial r} (4\pi r^2 \frac{\partial \psi}{\partial r})=-\frac{\hbar^{2}}{2m} \int_{0}^{\infty} 4\pi r^2dr \frac{1}{r^2} \psi^*\frac{\partial}{\partial r} ( r^2 \frac{\partial \psi}{\partial...
49. ### I Quantum wavefunction is real?

Last year I read the news article https://scitechdaily.com/physicists-prove-that-the-imaginary-part-of-quantum-mechanics-really-exists/ which says i is necessary for QM. You may be interested in it also.
50. ### Find g(x)/h(y) for a given F(x,y)

I agree with your minus result.