A proton is bound in a square well of width 4.0 fm = 4.0 ×10^-15 m. The depth of the well is six

A proton is bound in a square well of width 4.0 fm = 4.0...

Key Concepts

Finite Square Well Potential

A finite square well is a potential energy model in quantum mechanics where the particle is confined within a region of finite width with a constant potential, while the potential energy outside this region is higher. Unlike the infinite square well, the finite well allows for the possibility of tunneling, and its bound state energy levels are modified by the finite depth, requiring adjustment from the idealized infinite model for practical systems.

Infinite Potential Well

The infinite potential well is an idealized quantum mechanical model in which the particle is confined to a specific region with infinitely high potential barriers at the boundaries, leading to well-defined quantized energy states. These energy levels are determined analytically and serve as a reference for more complex potentials such as the finite square well, allowing comparison and estimation of energy shifts in more realistic scenarios.

Energy Quantization in Confined Systems

In quantum mechanics, particles confined within potential wells exhibit quantized energy levels due to the boundary conditions imposed on their wavefunctions. This quantization results in discrete allowed energy states that depend on the geometry and depth of the well, and is a central concept in understanding phenomena such as electronic transitions and spectral lines.

Energy Level Transitions

Transitions between quantized energy states in a confined system occur when a particle absorbs or emits energy, typically in the form of a photon. The energy of the photon corresponds to the difference between the initial and final energy levels, and these transitions are governed by selection rules that determine the possible transitions based on properties like angular momentum.

Photon Energy-Wavelength Relationship

The relationship between the energy of a photon and its wavelength is given by Planck’s relation, E = hc/?, where h is Planck’s constant and c is the speed of light. This fundamental equation links the quantized energy differences between states to the measurable wavelength of the photon involved in the absorption or emission process, thereby connecting the quantum mechanical behavior of the particle to observable phenomena.

Transcript

00:01 Okay, so problem number 25 states that a proton is bound in a square well of width four femtometers.

00:07 The depth of the well is six times the ground level energy of an electron in the ground state of a one -dimensional infinite square well.

00:14 The correspondence, yeah, one -demand, yeah, that was it was it.

00:17 If the proton makes a transition from the level with energy equals one to a level of energy equals three by absorbing a photon, find the wavelength of this photon.

00:25 All right.

00:25 So let's start with, so our potential is given by, u is equal to 6e1dw, which is the ground state energy of an electron in an infinite square well.

00:39 So this is given, we get the solutions to this is given to us in figure 40 .15b, where we get a square well, a finite square well, with the potential here, u0 is equal to 6e1dw.

01:01 Now it gives us the three energy levels of a particle in this particular box.

01:07 So for e3, that's going to be equal to 5 .09 times e1dw.

01:35 E1d .w.

01:36 Then down to about here, we get e2 is equal to.

01:42 2 .43 e1dw and then here e1 e1 is equal to 0 .625 e1dw.

01:59 So these are the corresponding energies of a particle in this particular box, in a box with this finite potential with this potential barrier.

02:14 Now, if a photon wants to, if a particle is going to be excited from the ground state up to the second excited state, the photon is going to come in with energy, called e gamma.

02:30 That energy has to be equal to this particular transition.

02:36 So this energy level, this depth here.

02:39 So in order to excite a particle from here to here, this photon has to be equal to that energy.

02:46 Now, so we need to know what that energy difference is.

02:51 So delta e is going to be equal to e3 minus e1, which is equal to 5 .09 minus 0 .625 times e1 in the one -dimensional well, which is just going to be equal to 4 .625.

03:15 5 .4 .465 times this energy of the one -dimensional well.

03:30 Now, so let's go ahead and solve for this energy.

03:36 So this energy is given by pi squared, h -bar squared, over 2ml squared.

03:47 So now these are all constants that we know...

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