Calculus Partial Derivatives Quiz

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Questions and Answers

What is the result of the partial derivative of the function $f = yx^2$ with respect to $y$?

$2yx$

$2xy$

$x^2$ (correct)

$y^2$

When calculating the partial derivative of $f = cx^2$ with respect to $x$, what is the result?

$cx$

$2c$

$c/x$

$2cx$ (correct)

In the expression for higher order derivatives, what occurs during the evaluation of second order partial derivatives?

The same variable is differentiated twice.

All variables are treated as constant in the second step.

Variables are ignored for the second calculation.

The derivative with respect to different variables is calculated sequentially. (correct)

Which equation is an example of where second order partial derivatives are commonly used?

Wave equation Signup and view all the answers

What is generally held constant when differentiating a function of multiple variables with respect to one variable?

All other variables Signup and view all the answers

What type of atoms does the Bohr theory apply to?

Hydrogen and one-electron ions Signup and view all the answers

What limitation of the Bohr theory relates to spectral line intensity?

It cannot explain why certain transitions occur more frequently Signup and view all the answers

Which scientific field emerged from the limitations of the Bohr theory?

Quantum mechanics Signup and view all the answers

What significant achievement is attributed to quantum mechanics by its early proponents?

Describing the physical world in a fundamentally new way Signup and view all the answers

What characteristic of quantum mechanics sets it apart from classical mechanics?

It describes systems at a microscopic level Signup and view all the answers

What was one of the key outcomes of applying quantum mechanics by the early 1930s?

Understanding many aspects of chemistry and physics Signup and view all the answers

Who were some of the pioneers in developing quantum mechanics?

Erwin Schrödinger, Werner Heisenberg, and Max Born Signup and view all the answers

What did Eugene Wigner refer to the discovery of quantum mechanics as?

A miracle Signup and view all the answers

What is the first step to find the second derivative of the function f with respect to x?

Calculate the first derivative of f with respect to x Signup and view all the answers

In the expression for the second derivative, what happens to the term containing y?

It is treated as a constant Signup and view all the answers

Which of the following represents a possible solution to the wave equation?

A combination of waves with different wavelengths and amplitudes Signup and view all the answers

In the wave equation y = A cos ω(t – x/v), what does 'A' represent?

The amplitude of the wave Signup and view all the answers

What type of wave is represented by y = A cos ω(t – x/v) if it is stationary?

A standing wave fixed at both ends Signup and view all the answers

What happens to the value of f when the second derivative with respect to y is calculated?

It yields a zero result independent of x Signup and view all the answers

Which of the following statements accurately describes a wave train?

It is a series of waves with a constant amplitude and wavelength. Signup and view all the answers

What role does 'v' play in the wave equation y = A cos ω(t – x/v)?

It represents the speed of wave propagation Signup and view all the answers

What is the second partial derivative of y with respect to x, according to the content?

$rac{∂^2y}{∂x^2}$ Signup and view all the answers

What is the significance of the wave function $ ilde{ ext{ψ}}$ in quantum mechanics?

It corresponds to the wave variable y. Signup and view all the answers

Which equation is identified as a solution of the wave equation?

Equation (5.5) Signup and view all the answers

What does the term 'complex' mean in the context of the wave function?

It involves real and imaginary components. Signup and view all the answers

When examining the wave function, what does holding x constant allow us to find?

The partial derivative of y with respect to t. Signup and view all the answers

Which of these statements is true about the second partial derivative with respect to t?

It is denoted as $rac{∂^2y}{∂t^2}$. Signup and view all the answers

What basic principle related to quantum mechanics cannot be derived from anything else?

A fundamental physical principle. Signup and view all the answers

In the context provided, what is the relationship between y and the wave equation?

y represents wave motion. Signup and view all the answers

What is the purpose of the trigonometric identity used in the context?

To simplify the integration process Signup and view all the answers

What variable do the given integrals rely on for normalization?

A Signup and view all the answers

Which expression corresponds to the probability of finding a particle in the interval between x and x + dx?

A^2 n^2 dx Signup and view all the answers

In the normalization process, what must the integral of the probability expression equal when properly normalized?

1 Signup and view all the answers

What does the term n represent in the equation given?

The quantum number Signup and view all the answers

Which of the following is an implication of normalizing a function?

The total probability becomes equal to unity Signup and view all the answers

What integral is performed to normalize the function?

∫ sin^2(nπx/L) dx Signup and view all the answers

What happens if the normalization condition is not satisfied?

The wave function becomes non-physical Signup and view all the answers

Why can't the expectation value for momentum be calculated in the same way as in classical physics?

The uncertainty principle prevents the specification of corresponding momentum when position is specified. Signup and view all the answers

What does applying Schrödinger’s equation to a particle's motion yield in quantum physics?

The wave function that describes the probabilities of its properties. Signup and view all the answers

What happens when classical physics is applied to the motion of a body under various forces?

The complete future course of the body’s motion can be predicted. Signup and view all the answers

What can be inferred about the expectation value for energy (E) in quantum mechanics?

E cannot be determined if the time is specified. Signup and view all the answers

What is the result of expressing the momentum as a function of position and time in quantum mechanics?

The uncertainty principle limits the specification of position and momentum simultaneously. Signup and view all the answers

Which statement best describes the role of operators in quantum mechanics?

Operators allow for the evaluation of expectation values in quantum systems. Signup and view all the answers

What does differentiating the free-particle wave function with respect to time indicate?

It provides insight into how momentum can be pulled from the wave function. Signup and view all the answers

How does classical physics treat the uncertainty principle compared to quantum physics?

The uncertainty principle is ignored in classical physics, leading to precise predictions. Signup and view all the answers

Study Notes

Chapter 5: Quantum Mechanics

Quantum mechanics is an approximation of classical mechanics
Quantum mechanics describes probabilities rather than certainties
The wave function Ψ, of a body, has no physical interpretation, but its absolute magnitude squared (|Ψ|²) is proportional to the probability of finding the body at a particular place and time
Wave functions are complex
The probability density |Ψ|² is Ψ*Ψ
Every acceptable wave function can be normalized
Wave function Ψ must be continuous and single-valued everywhere
Partial derivatives Ψ/∂x, Ψ/∂y, Ψ/∂z must be continuous and single-valued everywhere
Ψ must be normalizable (Ψ goes to 0 as x→∞, y→±∞, z → +∞ for the integral of Ψ² over all space to be a finite constant)
Classical mechanics is an approximation of quantum mechanics
The future history of a particle is completely determined by its initial position, momentum, and forces acting upon it (in classical mechanics)
Probabilities are related to cause and effect in quantum mechanics
For example, the radius of an electron's orbit in a ground state hydrogen atom is the most probable radius (5.3 x 10⁻¹¹m)

Schrödinger's Equation

Time-dependent form of Schrödinger's equation:

ih(dΨ/dt) = (h²/2m)(d²Ψ/dx²) + UΨ
Time-dependent Schrödinger equation in three dimensions:

ih(dΨ/dt)= (ħ²/2m)[(d²Ψ/dx²) + (d²Ψ/dy²) + (d²Ψ/dz²)] + UΨ
Schrödinger's equation can be derived from various methods but cannot be rigorously derived from existing physical principles
It is considered a basic principle
Schrödinger's equation can be used to find wave functions for a particle
Total energy of a particle is the sum of kinetic energy and potential energy
Schrödinger's time-dependent equation is not used when the system's potential energy does not depend on time explicitly

Particle in a Box

Particle in a rigid box, a theoretical model of atoms.
The particle is confined between x = 0 and x = L.
The particle's energy can have certain values.
The energy levels are quantized En = (n²h²)/(8mL²)
Wave functions that correspond to energy eigenvalues:

Ψn = A sin (nπx/L), normalized.
The particle is most likely to be found in the middle of the box (ground state n = 1), and least likely in other places.
This is a contrast to classical physics, which predicts equal probability anywhere in the box.

Finite Potential Well

The wave function penetrates the walls of the well.
Longer wavelengths and lower particle momenta; lower energy levels En
Boundary conditions (same value and slope at the edges of the well) determine the energy levels

Tunnel Effect

A particle with energy E < U (height of the barrier) can still have a probability of passing through the barrier.
The higher and wider the barrier, the smaller the probability
The tunnel effect has significant implications in radioactive decay and semiconductor devices
Transmission probability: T = exp(-2k₂L), where k₂= √2m(U-E)/h

Harmonic Oscillator

Harmonic motion is the vibration of a system about an equilibrium position
Restoring forces return the system to equilibrium
When amplitudes are small, systems behave like simple harmonic oscillators
Potential energy:

U = kx² (proportional to x²)
Energy levels: En = (n + ½)hv (equally spaced)
Lowest energy level = Zero-point energy (Eo= hv)
The spacing of the energy levels is constant for the harmonic oscillator.
Wave functions are not uniform; these have tails that extend into classically forbidden regions.

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Test your understanding of partial derivatives in multivariable calculus. This quiz covers basic calculations, the evaluation of higher-order derivatives, and common applications of second-order partial derivatives. Perfect for students looking to reinforce their knowledge in this area.