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andriy [413]
3 years ago
6

For a proton in the ground state of a 1-dimensional infinite square well, what is the probability of finding the proton in the c

entral 2% of the well?
Physics
1 answer:
Butoxors [25]3 years ago
5 0

Answer:

The probability of finding the proton at the central 2% of the well is almost exactly 4%

Explanation:

If we solve Schrödinger's equation for the infinite square well, we find that its eigenfunctions are sinusoidal functions, in particular, the ground state is a sinusoidal function for which only half a cycle fits inside the well.

let L be the well's length, the boundary conditions for the wavefunction are:

\psi(0) = \psi(L) =0

And Schrödinger's equation is:  

- \frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi

The solution to this equation are sines and cosines, but the boundary conditions only allow for sine waves. As we pointed out, the ground state is the sine wave with the largest wavelength possible (that is, with the smallest energy).

\psi_0(x)=\sqrt[]{\frac{2}{L} }\, \sin(\frac{\pi x}{L} )\\

here the leading constant is just there to normalise the wavefunction.

Now, if we know the wavefunction, we can know what the probability density function is, it is:

f_X(x) = |\psi|^2

So in our case:

f_X(x) = \frac{2}{L} \sin^2(\frac{\pi x}{L})

And to find the probability of finding the particle in a strip at the centre of the well of width 2% of L we only have to integrate:

P(X \in [0.49 L, 0.51L ])= \int\limits^{0.49L}_{0.51L } {\frac{2}{L} \sin^2(\frac{\pi x}{L})} \, dx

If we do a substitution:

x = u \, L

We get the integral:

\int\limits^{0.49}_{0.51 } 2\,  \sin^2(\pi u)} \, du

This integral can be computed analytically, and it's numerical value is .0399868, that is, almost a 4% probability.

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KatRina [158]

The velocity of the block when t == 2 s is 60.7 ft./sec.

Equations of Motion.

Here the friction is F_f = \mu_k N = 0.2 N

+ \uparrow \sum F_y = ma_y; \quad N – 10 = \frac { 10 } { 32.2 }(0) \quad N = 10 lb \\ \begin{aligned} \underrightarrow{ + } \sum F_x = ma_x; \quad 8t^2 – 0.2(10 &) = \frac { 10 } { 32.2 }a \\ & a = 3.22(8t^2 – 2) ft/s^2 \end{aligned}

Kinematics.

The velocity of the block as a function of t can be determined by

integrating dv = adt using the initial condition v = 4 ft./s at t = 0.

\int_{ 4 ft/s }^{ v } dv = \int_0^t 3.22(8t^2 – 2)dt \\ \begin{aligned} v – &4 = 3.22 (\frac 8 3 t^3 – 2t) \\ & v = \{8.5867t^3 – 6.44t + 4 \} ft/s \end{aligned}

The displacement as a function of t can be determined by integrating

ds = vdt using

the initial condition s = 0 at t = 0

\int_0^s ds = \int_0^t (8.5867t^3 – 6.44t + 4)dt \\ s = \{2.1467t^4 – 3.22t^2 + 4t \} ft

at t = 2 sec

s = 30 ft.

Thus, at s = 30 ft.,

\begin{aligned} v &= 8.5867(2.0089^3) – 6.44(2.0089) + 4 \\ &= 60.67 ft/s \\ &= 60.7 ft/s \end{aligned}

Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move.

Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system.

Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

Learn more about kinematics here : brainly.com/question/24486060

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Answer:

Equation for SHM can be written

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Answer:

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The kinetic energy is measured in Joules, which is a product of the mass of the substance and the time taken to travel a distance.

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