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Chapter 2: Problem 2

Show that \(E\) must exceed the minimum value of \(V(x)\), for every normalizable solution to the time-independent Schr枚dinger equation. What is the classical analog to this statement? Hint: Rewrite Equation \(2.5\) in the form $$ \frac{d^{2} \psi}{d x^{2}}=\frac{2 m}{\hbar^{2}}[V(x)-E] \psi: $$ if \(E

Short Answer

Expert verified
For bound states, \(E > V_{\min}\) ensures normalizability, and classically this means staying out of forbidden regions.

Step by step solution

01

Understand the Schr枚dinger Equation

The time-independent Schr枚dinger equation for a particle of mass \(m\) in a potential \(V(x)\) is given by:\[ -\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \]Rearrange it into the form:\[ \frac{d^2 \psi}{dx^2} = \frac{2m}{\hbar^2}[V(x) - E] \psi(x) \]This step sets the tone for comparing the potential \(V(x)\) with the energy \(E\).
02

Analyze the Differential Equation

The equation \( \frac{d^2 \psi}{dx^2} = \frac{2m}{\hbar^2}[V(x)-E] \psi(x) \) indicates that the sign of \( \frac{d^2 \psi}{dx^2} \) is determined by \([V(x) - E] \psi(x)\). If \(E < V_{\min}\), then \(V(x) - E > 0\) for all \(x\), implying that \(\psi\) and \(\frac{d^2 \psi}{dx^2}\) have the same sign across its domain.
03

Implication on Normalizability

If \(\psi\) and its second derivative always have the same sign, \(\psi\) would either diverge to \(\infty\) or \(-\infty\) as \(x\) approaches infinity. Such behavior implies that \(\psi\) cannot be normalized because the integral \(\int_{-\infty}^{\infty} |\psi(x)|^2 dx\) would diverge, which contradicts the requirement for a quantum mechanical wave function to be normalizable.
04

Connect to Classical Analogy

In classical mechanics, for a particle to be bound, its energy \(E\) must be greater than the potential energy \(V(x)\) at all accessible positions (i.e., avoiding infinite regions where \(E<V(x)\)), similar to requiring \(E\) to be greater than \(V_{\min}\) for a bound quantum state. This ensures that the particle's total energy allows for kinetic movement rather than being confined to turning points in potential wells.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that explains the behavior and interactions of matter and energy at atomic and subatomic levels. It describes how particles such as electrons and photons exist in multiple states simultaneously until they are observed or measured. In quantum mechanics, the Schr枚dinger Equation is crucial as it dictates how quantum systems evolve over time. This equation allows us to determine the wave function, which provides information about the probabilities of a particle's position and momentum.
  • The Schr枚dinger Equation is a type of differential equation that relates the energy of a particle to its wave function.
  • It allows physicists to predict future behavior and properties of quantum systems, given initial conditions.
Understanding quantum mechanics requires a shift from classical physics perspectives, such as determinism, to acknowledge the probabilistic nature of quantum realities.
Wave Function Normalization
Wave function normalization is an essential concept in quantum mechanics as it ensures that the total probability of finding a particle within all space equals one. The wave function, denoted as \( \psi(x) \), represents the probability amplitude of a particle's position. However, to extract meaningful probabilities, the wave function must be normalized.
  • Normalization involves adjusting \( \psi(x) \) so that the integral over all space of the probability density \( |\psi(x)|^2 \) equals one.
  • Mathematically, this is expressed as \( \int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1 \).
In scenarios where a wave function cannot be normalized, it often indicates physical inconsistencies, such as the aforementioned problem where \( E < V_{\min} \), causing \( \psi(x) \) to diverge. Proper normalization remains fundamental for ensuring the validity of quantum mechanical predictions.
Potential Energy
In quantum mechanics, potential energy \( V(x) \) plays a central role in shaping the potential landscape that a particle "feels" as it moves. It influences the possible states a quantum particle can occupy and dictates how the wave function evolves under the Schr枚dinger Equation.
  • Potential energy often represents interactions such as electromagnetic forces or gravitational fields acting on the particle.
  • For a particle to be bound, its total energy \( E \) must exceed \( V_{\min} \), the minimum of the potential energy within its region.
In classical terms, this requirement ensures a particle has enough kinetic energy to avoid being trapped at a maximum potential energy point. Quantum mechanically, it means the wave function remains confined and normalizable, leading to stable and physically valid solutions.

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Most popular questions from this chapter

In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region? Hint: Classically, the energy of an oscillator is \(E=(1 / 2) k a^{2}=\) \((1 / 2) m \omega^{2} a^{2}\), where \(a\) is the amplitude. So the "classically allowed region" for an oscillator of energy \(E\) extends from \(-\sqrt{2 E / m \omega^{2}}\) to \(+\sqrt{2 E / m \omega^{2}}\). Look in a math table under "Normal Distribution" or "Error Function" for the numerical value of the integral.

Show that \(\left[A e^{i k x}+B e^{-i k x}\right]\) and \([C \cos k x+D \sin k x]\) are equivalent ways of writing the same function of \(x\), and determine the constants \(C\) and \(D\) in terms of \(A\) and \(B\), and vice versa. Comment: In quantum mechanics, when \(V=0\), the exponentials represent traveling waves, and are most convenient in discussing the free particle, whereas sines and cosines correspond to standing waves, which arise naturally in the case of the infinite square well.

What is the Fourier transform of \(\delta(x)\) ? Using Plancherel's theorem, show that $$ \delta(x)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} e^{i k x} d k $$ Comment: This formula gives any respectable mathematician apoplexy. Although the integral is clearly infinite when \(x=0\), it doesn't converge (to zero or anything else) when \(x \neq 0\), since the integrand oscillates forever. There are ways to patch it up (for instance, you can integrate from \(-L\) to \(+L\), and interpret Equation \(2.144\) to mean the average value of the finite integral, as \(L \rightarrow \infty\) ). The source of the problem is that the delta function doesn't meet the requirement (square-integrability) for Plancherel's theorem (see footnote 33). In spite of this, Equation \(2.144\) can be extremely useful, if handled with care.

Consider the potential $$ V(x)=-\frac{\hbar^{2} a^{2}}{m} \operatorname{sech}^{2}(a x) $$ where \(a\) is a positive constant, and "sech" stands for the hyperbolic secant. (a) Graph this potential. (b) Check that this potential has the ground state $$ \psi_{0}(x)=A \operatorname{sech}(a x) $$ and find its energy. Normalize \(\psi_{0}\), and sketch its graph (c) Show that the function $$ \psi_{k}(x)=A\left(\frac{i k-a \tanh (a x)}{i k+a}\right) e^{i k x} $$ (where \(k \equiv \sqrt{2 m E} / \hbar\), as usual) solves the Schr枚dinger equation for any (positive) energy \(E\). Since \(\tanh z \rightarrow-1\) as \(z \rightarrow-\infty\), $$ \psi_{k}(x) \approx A e^{i k x} . \quad \text { for large negative } x . $$ This represents, then, a wave coming in from the left with no accompanying reflected wave (i.e., no term \(\exp (-i k x)\) ). What is the asymptotic form of \(\psi_{k}(x)\) at large positive \(x ?\) What are \(R\) and \(T\), for this potential? Comment: This is a famous example of a reflectionless potential- every incident particle, regardless of its energy, passes right through. \({ }^{47}\)

*Problem 2.7 A particle in the infinite square well has the initial wave function 15 $$ \Psi(x, 0)=\left\\{\begin{array}{ll} A x, & 0 \leq x \leq a / 2 \\ A(a-x) . & a / 2 \leq x \leq a (c) What is the probability that a measurement of the energy would yield the value \(E_{1}\) ? (d) Find the expectation value of the energy. \end{array}\right. $$ (a) Sketch \(\Psi(x, 0)\), and determine the constant \(A\). (b) Find \(\Psi(x, t)\).
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