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- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/06%3A_Scattering_from_Potential_Steps_and_Square_Barriers
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/03%3A_The_Schrodinger_Equation/3.03%3A_Analysis_of_the_wave_equationOne of the important aspects of the Schrödinger equation(s) is its linearity. For the time independent Schrödinger equation, which is usually called an eigenvalue problem.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/05%3A_Innite_Wells/5.01%3A_Zero_of_Energy_is_ArbitraryThat is a very workable definition, except in one case: if we take a square well and make it deeper and deeper, the energy of the lowest state decreases with the bottom of the well. As the well depth g...That is a very workable definition, except in one case: if we take a square well and make it deeper and deeper, the energy of the lowest state decreases with the bottom of the well. As the well depth goes to infinity, the energy of the lowest bound state reaches −∞, and so does the second, third etc. It makes much more physical sense to define the bottom of the well to have zero energy, and the potential outside to have value V 0, which goes to infinity.
- https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Nuclear_and_Particle_Physics_(Walet)/09%3A_Relativistic_Kinematics/9.01%3A_Lorentz_Transformations_of_Energy_and_MomentumFrom the Lorentz transformation property of time and position, for a change of velocity along the \(x\)-axis from a coordinate system at rest to one that is moving with velocity \({\vec{v}} = (v_x,0,0...From the Lorentz transformation property of time and position, for a change of velocity along the \(x\)-axis from a coordinate system at rest to one that is moving with velocity \({\vec{v}} = (v_x,0,0)\) we have We know however that the full four-momentum is conserved, i.e., if we have two particles coming into a collision and two coming out, the sum of four-momenta before and after is equal,
- https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Nuclear_and_Particle_Physics_(Walet)/08%3A_Symmetries_of_the_theory_of_strong_interactions/8.04%3A__SU(4)%2C_SU(5)%2C_and_SU(6)_flavor_symmetriesOnce we have three flavors of quarks, we can ask the question whether more flavors exists. At the moment we know of three generations of quarks, corresponding to three generations (pairs).
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/12%3A_Quantum_Mechanics_of_the_Hydrogen_Atom/12.05%3A_Smaller_Effects/12.5.03%3A_The_Zeeman_EffectIncluding hyperfine structure with the Zeeman effect is more difficult, since the field associated with the proton magnetic dipole moment is weak, and hence it does not take a particularly strong exte...Including hyperfine structure with the Zeeman effect is more difficult, since the field associated with the proton magnetic dipole moment is weak, and hence it does not take a particularly strong external field to make the Zeeman effect comparable in magnitude to the strength of the hyperfine interactions.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/08%3A_The_Formalism_Underlying_Quantum_Mechanics/8.03%3A_The_Measurement_ProcessIf we measure \(E\) once and we find \(E_i\) as outcome we know that the system is in the \(i\) th eigenstate of the Hamiltonian. This is called the "collapse of the wave function": before the first m...If we measure \(E\) once and we find \(E_i\) as outcome we know that the system is in the \(i\) th eigenstate of the Hamiltonian. This is called the "collapse of the wave function": before the first measurement we couldn't predict the outcome of the experiment, but the first measurements prepares the wave function of the system in one particuliar state, and there is only one component left!
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/10%3A_Time-Dependent_Wavefunctions/10.03%3A_Completeness_and_time-dependenceIn the discussion on formal aspects of quantum mechanics I have shown that the eigenfunctions to the Hamiltonian are complete, i.e., for any \(\psi(x, t)\) \[\psi(x, t)=\sum_{n=1}^{\infty} c_n(0) e^{-...In the discussion on formal aspects of quantum mechanics I have shown that the eigenfunctions to the Hamiltonian are complete, i.e., for any \(\psi(x, t)\) \[\psi(x, t)=\sum_{n=1}^{\infty} c_n(0) e^{-i E t / \hbar} \phi_n(x),\] so that the time dependence is completely fixed by knowing \(c(0)\) at time \(t=0\) only! In other words if we know how the wave function at time \(t=0\) can be written as a sum over eigenfunctions of the Hamiltonian, we can then determibe the wave function for all times.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/11%3A_3D_Schrodinger_Equation/11.05%3A_Now_where_does_the_probability_peakbut what is the probability to find the electron at a distance \(r\) from the proton? The key point to realise is that for each value of \(r\) the electron can be anywhere on the surface of a sphere o...but what is the probability to find the electron at a distance \(r\) from the proton? The key point to realise is that for each value of \(r\) the electron can be anywhere on the surface of a sphere of radius \(r\), so that for larger \(r\) more points contribute than for smaller \(r\). \[\frac{d}{d r} P_1=\frac{4}{a_0^3}\left(2 r e^{-2 r / a_0}-2 r^2 / a_0 e^{-2 r / a_0}\right)\]
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/12%3A_Quantum_Mechanics_of_the_Hydrogen_Atom/12.03%3A_Schrodinger_Theory_of_the_Hydrogen_Atom/12.3.01%3A_Schrodinger_Theory_of_Hydrogenwhere the first term of the left represents the particle's kinetic energy, the second the particle's potential energy, and H is called the Hamiltonian of the system. The upshot is, with the solution w...where the first term of the left represents the particle's kinetic energy, the second the particle's potential energy, and H is called the Hamiltonian of the system. The upshot is, with the solution written as it is here, that the numbers \(n, l\), and \(m_l\), called quantum numbers of the electron, can only take on particular integer values, and each of these corresponds to the quantization of some physical quantity.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Walet)/07%3A_The_Harmonic_Oscillator/7.02%3A_Dimensionless_Coordinates\[E=\frac{1}{2} m \bar{x}^2+\frac{1}{2} m \omega^2 x^2=\frac{1}{2 m} p^2+\frac{1}{2} m \omega^2 x^2\] \[\widehat{H}=\frac{1}{2 m} \frac{1}{2m}\widehat{p}^2+\frac{1}{2} m \omega^2 x^2=-\frac{\hbar^2}{2...\[E=\frac{1}{2} m \bar{x}^2+\frac{1}{2} m \omega^2 x^2=\frac{1}{2 m} p^2+\frac{1}{2} m \omega^2 x^2\] \[\widehat{H}=\frac{1}{2 m} \frac{1}{2m}\widehat{p}^2+\frac{1}{2} m \omega^2 x^2=-\frac{\hbar^2}{2 m} \frac{d^2}{d x^2}+\frac{1}{2} m \omega^2 x^2 .\] \[-\frac{\hbar^2}{2 m} \frac{m \omega}{\hbar} \frac{d^2}{d y^2} u(y)+\frac{1}{2} m \omega^2 \frac{\hbar}{m \omega} y^2 u(y)=\epsilon \hbar \omega u(y)\]
