Derive the formula for kinetic energy of a particle having mass m and velocity v using dimensional analysis

Learning Objectives By the end of this section, you will be able to: • Calculate the kinetic energy of a particle given its mass and its velocity or momentum • Evaluate the kinetic energy of a body, relative to different frames of reference It’s plausible to suppose that the greater the velocity of a body, the greater effect it could have on other bodies. This does not depend on the direction of the velocity, only its magnitude. At the end of the seventeenth century, a quantity was introduced into mechanics to explain collisions between two perfectly elastic bodies, in which one body makes a head-on collision with an identical body at rest. The first body stops, and the second body moves off with the initial velocity of the first body. (If you have ever played billiards or croquet, or seen a model of Newton’s Cradle, you have observed this type of collision.) The idea behind this quantity was related to the forces acting on a body and was referred to as “the energy of motion.” Later on, during the eighteenth century, the name kinetic energy was given to energy of motion. With this history in mind, we can now state the classical definition of kinetic energy. Note that when we say “classical,” we mean non-relativistic, that is, at speeds much less that the speed of light. At speeds comparable to the speed of light, the special theory of relativity requires a different expression for the kinetic energy of a particle, as discussed in Since objects (or systems) of interest vary...

\( \newcommand\): To the left the wavefunction, to the right a representation of the probability of finding the particle at a specific position for the various quantum states. Additionally, since the probability of finding the particle somewhere in the well is equal to 1, and the probability of finding the particle is equal to the wavefunction squared, using an integral table yields the result This result has a number of extremely important features. • The particle can only have certain, discrete values for energy. In classical physics, a particle trapped in a region of space can have any, continuous value for energy. The restriction of a bound particle to specific, quantized values of allowed energy is a hallmark of quantum mechanics. • The particle cannot be at rest. Notice that the lowest possible energy for the particle is in the \(n = 1\) state, which has non-zero energy. This is termed the zero-point energy, and can be understood as a consequence of the Heisenberg uncertainty principle. This is in complete contrast to classical physics. • The particle has regions of high and low probability of being found in the well. The probability of finding the particle is equal to the square of the wavefunction, which is not a constant value. Unlike classical physics, where the particle is equally likely to be anywhere in the well, in quantum mechanics there exist positions where the particle will never be found, and regions where the probability of finding the particle is great...

5 Relativity • Introduction • 5.1 Invariance of Physical Laws • 5.2 Relativity of Simultaneity • 5.3 Time Dilation • 5.4 Length Contraction • 5.5 The Lorentz Transformation • 5.6 Relativistic Velocity Transformation • 5.7 Doppler Effect for Light • 5.8 Relativistic Momentum • 5.9 Relativistic Energy • Learning Objectives By the end of this section, you will be able to: • Describe how to set up a boundary-value problem for the stationary Schrӧdinger equation • Explain why the energy of a quantum particle in a box is quantized • Describe the physical meaning of stationary solutions to Schrӧdinger’s equation and the connection of these solutions with time-dependent quantum states • Explain the physical meaning of Bohr’s correspondence principle In this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energy of the particle is quantized as a consequence of a standing wave condition inside the box. Consider a particle of mass m m that is allowed to move only along the x-direction and its motion is confined to the region between hard and rigid walls located at x = 0 x = 0 and at x = L x = L ( infinite square well, described by the potential energy function 7.32 where E is the total energy of the particle. What types of solutions do we expect? The energy of the particle is a positive number, so if the value of the wave function is positive (rig...