# Classical Mechanics Solved Problems Classically, $F = mg$ and $g \approx 10 \frac$, so the ball feels a force of \, \mathrm$. And it turns out that $$\mathrm \,\varrho_(t,x_0,\xi_0)\hat_= S(t,x_0,\xi_0) \mathrm(\hslash)\; ,$$ where$S(t,x_0,\xi_0)$is the of the particle at time$t$, corresponding to initial condition$(x_0,\xi_0)$.To see that the peak in the wavefunction obeys Newton's laws, you can appeal to Ehrenfest's theorem, $$m \frac = - \left\langle \frac \right\rangle$$ which immediately gives that result.You may still be troubled, because in classical mechanics we need to specify an initial position and initial velocity, while in quantum mechanics it seems we only need to specify the analogue of position. The "velocity" of a particle is encoded by how fast the phase winds around in position.So we start with the 3-dimensional time-independent Schrodinger equation with a potential of$V = -GM/r$: $$-\frac\nabla^2 \psi - \frac\psi = E\psi \,.$$ This is effectively the same equation as for solving the Hydrogen atom: a And it turns out that $$\mathrm \,\varrho_(t,x_0,\xi_0)\hat_= S(t,x_0,\xi_0) \mathrm(\hslash)\; ,$$ where$S(t,x_0,\xi_0)$is the of the particle at time$t$, corresponding to initial condition$(x_0,\xi_0)$. To see that the peak in the wavefunction obeys Newton's laws, you can appeal to Ehrenfest's theorem, $$m \frac = - \left\langle \frac \right\rangle$$ which immediately gives that result. You may still be troubled, because in classical mechanics we need to specify an initial position and initial velocity, while in quantum mechanics it seems we only need to specify the analogue of position. The "velocity" of a particle is encoded by how fast the phase winds around in position. So we start with the 3-dimensional time-independent Schrodinger equation with a potential of$V = -GM/r$: $$-\frac\nabla^2 \psi - \frac\psi = E\psi \,.$$ This is effectively the same equation as for solving the Hydrogen atom: a$1/r$potential but with different coefficients. So the answer should take the same form, and we get all the energy levels, the n/l/m quantum numbers, etc. || And it turns out that $$\mathrm \,\varrho_(t,x_0,\xi_0)\hat_= S(t,x_0,\xi_0) \mathrm(\hslash)\; ,$$ where$S(t,x_0,\xi_0)$is the of the particle at time$t$, corresponding to initial condition$(x_0,\xi_0)$.To see that the peak in the wavefunction obeys Newton's laws, you can appeal to Ehrenfest's theorem, $$m \frac = - \left\langle \frac \right\rangle$$ which immediately gives that result.You may still be troubled, because in classical mechanics we need to specify an initial position and initial velocity, while in quantum mechanics it seems we only need to specify the analogue of position. The "velocity" of a particle is encoded by how fast the phase winds around in position.So we start with the 3-dimensional time-independent Schrodinger equation with a potential of$V = -GM/r$: $$-\frac\nabla^2 \psi - \frac\psi = E\psi \,.$$ This is effectively the same equation as for solving the Hydrogen atom: a$1/r$potential but with different coefficients.So the answer should take the same form, and we get all the energy levels, the n/l/m quantum numbers, etc.That is to say, after a generic interaction with the environment the ball will quickly end up with its position peaked about a narrow value.(Not necessarily so narrow that the uncertainty principle comes into play, but effectively zero for all macroscopic purposes.) Such states exist in the Hilbert space, as complicated linear combinations of the energy eigenstates.The average position of the particle is then given by $$\mathrm \,\varrho_(t,x_0,\xi)\,\hat_\; ,$$ where$\hat_$is the position operator (I have put the$\hslash$-dependence on the position operator because in general quantum operators depend on$\hslash$, however in the standard QM representation of the canonical commutation relations the position operator is independent of$\hslash$, and all the dependence is on the momentum operator; one could change this by means of a unitary transformation).The function$\mathrm \,\varrho_(t,x_0,\xi_0)\hat_$is a function of time$t$, of position$x_0$, momentum$\xi_0$through the initial quantum condition$\varrho_(x_0,\xi_0)$, and of$\hslash$.It would occupy some standing wave about the Earth and not evolve in time.The reason we never see macroscopic objects in such states is because they are unstable in the same sense as Schrodinger's cat. /r$ potential but with different coefficients.So the answer should take the same form, and we get all the energy levels, the n/l/m quantum numbers, etc.That is to say, after a generic interaction with the environment the ball will quickly end up with its position peaked about a narrow value.(Not necessarily so narrow that the uncertainty principle comes into play, but effectively zero for all macroscopic purposes.) Such states exist in the Hilbert space, as complicated linear combinations of the energy eigenstates.The average position of the particle is then given by $$\mathrm \,\varrho_(t,x_0,\xi)\,\hat_\; ,$$ where $\hat_$ is the position operator (I have put the $\hslash$-dependence on the position operator because in general quantum operators depend on $\hslash$, however in the standard QM representation of the canonical commutation relations the position operator is independent of $\hslash$, and all the dependence is on the momentum operator; one could change this by means of a unitary transformation).The function $\mathrm \,\varrho_(t,x_0,\xi_0)\hat_$ is a function of time $t$, of position $x_0$, momentum $\xi_0$ through the initial quantum condition $\varrho_(x_0,\xi_0)$, and of $\hslash$.It would occupy some standing wave about the Earth and not evolve in time.The reason we never see macroscopic objects in such states is because they are unstable in the same sense as Schrodinger's cat.

## Comments Classical Mechanics Solved Problems

• ###### Solved Problems in Classical Physics - bayanbox.ir
Reply

Preface This book complements the book 1000 Solved Problems in Modern Physics by the same author and published by Springer-Verlag so that bulk of the courses for undergraduate curriculum are covered.…

• ###### Classical Mechanics - Live Physics
Reply

Classical Mechanics. Projectile Motion – Fired at ground level A football is kicked with an initial velocity of 25 m/s at an angle of 45-degrees with the horizontal. Determine the time of flight, the horizontal displacement, and the peak height of the football. Projectile Motion – Pool ball leaves the table with initial horizontal velocity A.…

• ###### Mathematical methods of classical mechanics-Arnold V. I.
Reply

Of celestial mechanics, connected with the requirements of space exploration, created new interest in the methods and problems of analytical dynamics. The connections between classical mechanics and other areas of mathe­ matics and physics are many and varied. The appendices to this book are devoted to a few of these connections.…

• ###### Solved Problems in Classical Mechanics, Analytical and.
Reply

Booktopia has Solved Problems in Classical Mechanics, Analytical and Numerical Solutions with Comments by O. L. de Lange.…

• ###### Solved Problems in Classical Mechanics - Hardcover - Owen de.
Reply

Solved Problems in Classical Mechanics Analytical and Numerical Solutions with Comments Owen de Lange and John Pierrus. Provides a wide-ranging set of problems, solutions and comments in classical mechanics. Ranges from second-year level to postgraduate; Designed to develop problem-solving abilities…

• ###### Solved Problems in Classical Mechanics - پرشین‌گیگ
Reply

Solved Problems in Classical Mechanics vt= drt dt, 1 and the acceleration at, which is the time rate of change of the velocity, at= dvt dt. 2 It follows from 1 and 2 that the acceleration is also the second derivative a= d2r dt2. 3 Sometimes use is made of Newton’s notation, where a dot denotes diﬀerentiation with…

• ###### Lecture Notes on Classical Mechanics A Work in Progress
Reply

Lecture Notes on Classical Mechanics A Work in Progress Daniel Arovas Department of Physics University of California, San Diego May 8, 2013…

• ###### What are the best books for solutions of problems on.
Reply

Well, I am assuming that you want to study classical mechanics and want to understand every concept from the grass root level then “cengage mechanics part 1 and 2.…

• ###### Solved Problems in Classical Mechanics Analytical and.
Reply

Solved Problems in Classical Mechanics Analytical and Numerical Solutions with Comments. These include one-, two-, and three- dimensional motion; linear and nonlinear oscillations; energy, potentials, momentum, and angular momentum; spherically symmetric potentials; multi-particle systems; rigid bodies; translation and rotation.…

• ###### Collection of Problems in Classical Mechanics ScienceDirect
Reply

Collection of Problems in Classical Mechanics presents a set of problems and solutions in physics, particularly those involving mechanics. The coverage of the book includes 13 topics relevant to classical mechanics, such as integration of one-dimensional equations of motion; the Hamiltonian equations of motion; and adiabatic invariants.…