Quantum Mechanics: Physical Problems in One-Dimension

We start our escapade of solving one-dimensional physical problems with the Free Particle, one that is easy to solve by hard to understand. Solving the Schrödinger gives us continuous states.

After solving for the Free Particle, we have a brief discussion of the solutions and, by using the superposition principle along with the concept of a Fourier Transfer, the idea of localizing a wave function.

In this lesson, we ask the fundamental question: Are the wave solutions to the free particle physical, as in can it be used to represent a particle in the real world?

With our aim of localizing the wave function, we use a Gaussian wave packet in the form of φ(k) for our Fourier Transform. The values of the function φ(k) will be used as the amplitudes of the solutions that when superimposed, interfere constructively in a finite region.

Now comes the rigorous calculations involving difficult integrals to get our end product - a wave packet with wave number kâ

This system involves a particle confine to a limited region whereby bound states occurs. We see, for the first time, the process of deriving the discrete energy values for certain systems, i.e. a hydrogen atom. The energy is quantized.

Once we have determine the energy values, notice that n=0 gives E_o=0, an interesting result indeed. We shall soon see that quantum systems cannot have zero energy and so we label E₁ as the ground state energy or zero-point energy.

The Potential Barrier, case: E ≥ V_o

We easily infer the quantum mechanical study from the treatment of the potential step. For this scattering problem where the energy of the the particle is greater than the potential, we simply solve the Schrödinger equation keeping in mind that the potential is V_o only for the width of the barrier.

In analyzing the situation, we need the transmission and reflection coefficients. Apply the continuity conditions at the points x = 0 and x = a to solve the various intensities of the waves and thereby calculating R and T.



Beware, some heavy rearranging of algebraic expressions ahead. And remember that R + T = 1.

The Potential Barrier, case: E ≥ V_o

Now, with an algebraic expression for the transmission coefficient, we adjust energy values and observe what happens to the particle.



1. With E â

The Potential Barrier, case: E ≥ V_o

We wrap up by again investigating on the different effects of changing the parameters of the potential barrier but this time, we'll adjust the width of the barrier.



Applications to physics and engineering problems (towards the end): We manipulate the parameters that are given to us, width of barrier, energy of particle, potential to achieve our desired result.

The Potential Barrier, case: E ≤ V_o - Tunneling

We now face the most common problem in Quantum mechanics when dealing with physical problems - the phenomena of tunneling, how a particle can tunnel through a classical impenetrable barrier. This lesson sets out the situation looking at the Schrodinger equation.



Note: Unlike the previous problems, I hope you watch all of the first four of this series, Lec 20 to Lec 24, to fully understand this phenomena.

The Potential Barrier, case: E ≤ V_o - Tunneling

With rigorous calculations, we finally see the fruits of our work. With irrefutable evidence, we explain conclusive with a non-zero probability from the transmission coefficient that quantum mechanical objects can tunnel through classically impenetrable barriers, very unlike classical mechanics.



Note: Unlike the previous problems, I hope you watch all of the first four of this series, Lec 20 to Lec 23, to fully understand this phenomena.

The Potential Barrier, case: E ≤ V_o - Tunneling

Our last step of analyzing the idea of tunneling is to investigate the probability or proportion of particles that will tunnel through the potential barrier upon adjusting the various parameters.



Note: Unlike the previous problems, I hope you watch all of the first four of this series, Lec 20 to Lec 23, to fully understand this phenomena.

The Potential Barrier, case: E ≤ V_o - Tunneling

To conclude our study of tunneling, we glimpse at an approximation method we can use to calculate the probability of a particle tunneling through a potential we might encounter in the real world - where potentials are not simple square barriers.

The Finite Square Well Potential, case: E ≥ V_o



Our final problem before we head of to the harmonic oscillator. For the finite square well potential, the two interesting cases are when E ≥ V_o and E ≤ V_o. We look at the case where E ≥ V_o in which we expect a continuous doubly-degenerate energy spectrum.



Contrast results with classical mechanics where there is full transmission.

The Finite Square Well Potential, case: E ≥ V_o

We venture back again to the fundamental question of whether the wave solutions to the Schrödinger are physical. We sketch the probabilities densities |Ψ|² and again see that they have no physical meaning - it is absurd to say that a particle can be found in a region with probability of say 4.

The Finite Square Well Potential, case: E ≥ V_o

We remedy the problem of unphysical solutions with a Fourier transform, just like how we did it for the free particle. Just that now, the situation is a little more involved where we need to consider three regions separately.

The Finite Square Well Potential, case: E ≤ V_o

Classically, when E ≤ V the particle is confined in the well. It will bounce back and forth with constant momentum. Quantum mechanically, we expect solutions to yield a discrete energy spectrum and wave functions that decay in the regions outside a well.

The Finite Square Well Potential, case: E ≤ V_o

Solving the Schrödinger equation inside the well is a little different. Recall the theorem that bound states of a particle in a symmetric potential have definite parity: they are either even or odd functions.



V(-x) = V(x) ⇒ Ψ(-x) = ±Ψ(x)



We use this theorem to progress in the problem.

The Finite Square Well Potential, case: E ≤ V_o

To accomplish our objective or finding the energy values, we apply the boundary conditions, but this time we will apply it to both the antisymmetric and symmetric solutions at only ONE point, that is x = -a/2 or x = a/2.



Application of such conditions varies based on what we are finding. Here, it is the energy values and not the transmission coefficients.

The Finite Square Well Potential, case: E ≤ V_o

In graphing the expressions for each equation, we can this famous graph in quantum mechanics showing the discrete energy values for a finite square well potential.



Note that the number of solutions depend on the size of R, which in turn depends on the depth V_o and the width a of the well.



Also, in the limiting case V_o→∞, the circle's radius R becomes infinite we recover the energy expression for the infinite well.