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Chapter 5: Problem 5

Just what is stationary in a stationary state? The particle? Something else?

Short Answer

Expert verified
In a 'stationary state', what stays stationary isn't the particle but rather the probability distribution of its properties like energy, position and others that commute with Hamiltonian. This means, the properties derived from the wave function do not change with time. So 'stationary' here does not denote a lack of motion but rather a state where key particle parameters remain invariant over time.

Step by step solution

01

Understanding Stationary State

In the realm of quantum mechanics, when we discuss a system being in a 'stationary state', we are not referring to a lack of motion. In fact, the particle in question can very much be moving.
02

Concept and Characteristics of Stationary State

The term 'stationary state' is used to describe situations where specific properties of a system do not change in respect to time. For example, the system's energy, and all other observable quantities that commute with Hamiltonian do not change with time. This is due to these observable quantities being derived from the wave function, which itself does not change over time in a stationary state.
03

Explaining Particle's Behaviour

The particle, in a stationary state, can be associated with some probability distribution of locating it at different positions, which does not change with time. Even though a particle's specific position or momentum isn't defined in quantum mechanics, the probability distribution of these variables remains constant in a stationary state.

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Key Concepts

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

Stationary State
In quantum mechanics, the notion of a "stationary state" can seem confusing at first. Unlike the word "stationary" which might imply stillness, it actually refers to specific parameters of a system remaining constant over time. In other words, it's not the particle itself that is stationary.
A stationary state is defined by its constant energy level, meaning the energy does not vary as time progresses. This type of state can help physicists predict behaviors without daily changes in certain measurable properties.
Importantly, in a stationary state, while the particle's energy is static, its motion may not be. The term specifically denotes the stability of observable quantities, like energy, that commute with the Hamiltonian.
This allows certain predictions about the particle's behavior, such as knowing that its probability distribution 鈥� a key concept in quantum mechanics 鈥� doesn't shift over time.
Wave Function
Wave functions serve as a backbone in quantum mechanics by providing insight into the probable location and characteristics of particles. When we talk about a wave function in a stationary state, it's crucial to understand that it does not change over time.
Wave functions are mathematical expressions that encompass all the possible information about a quantum system. For a stationary state, the wave function presents a continuous probability distribution, highlighting where a particle might be at any given moment.
  • They encapsulate probabilities instead of definite outcomes.
  • Describe a wide range of possible states for particles.
In stationary states, since these functions don鈥檛 evolve with time, observers see steady probabilities of where the particles could be, even though they can't pinpoint exact locations. This characteristic is opposite to non-stationary states, where wave functions are time-dependent and probabilities shift as time flows.
Hamiltonian
The Hamiltonian is a fundamental concept in quantum mechanics, representing the total energy of a system 鈥� typically including both kinetic and potential energy. It plays a pivotal role in defining stationary states.
The Hamiltonian operator is partly responsible for determining the stationary nature of certain observable quantities in quantum systems. When a system is in a stationary state, the Hamiltonian, which is a constant, determines that these quantities do not change over time.
  • Stationary states are eigenstates of the Hamiltonian.
  • Working with Hamiltonians helps predict the time evolution of quantum systems.
By commuting with the Hamiltonian, observable properties like energy remain unchanged, thus calling upon the Hamiltonian provides a consistent framework for understanding a system's potential behaviors without surprises over time.
The Hamiltonian's role is crucial because it bridges time-dependent and time-independent Schr枚dinger equations, linking classical mechanics' energy concepts with quantum behaviors.

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Most popular questions from this chapter

Consider the differential equation \(d^{2} f(x) / d x^{2}=b f(x)\). (a) Suppose that \(f_{1}(x)\) and \(f_{2}(x)\) are solutions. That is, $$ \frac{d^{2} f_{1}(x)}{d x^{2}}=b f_{1}(x) \text { and } \frac{d^{2} f_{2}(x)}{d x^{2}}=b f_{2}(x) $$ Show that the equation also holds when the linear combination \(A_{1} f_{1}(x)+A_{2} f_{2}(x)\) is inserted. (b) Suppose that \(f_{3}(x)\) and \(f_{4}(x)\) are solutions of \(d^{2} f(x) / d x^{2}=b f^{2}(x)\). Is \(A_{3} f_{3}(x)+A_{4} f_{4}(x)\) a solution? Justify your answer.

Consider a particle in the ground state of a finite well. Describe the changes in its wave function and energy as the walls are made progressively higher \(\left(U_{0}\right.\) is increased) until essentially infinite.

Classically, if a particle is not observed, the probability per unit length of finding it in a box is a constant \(1 / L\) along the entire length of the box. With this, show that the classical expectation value of the position is \(\frac{1}{2} L\), that the expectation value of the square of the position is \(\frac{1}{3} L^{2}\). and that the uncertainty in position is \(L / \sqrt{12}\).

Exercises \(94-97\) refer to a bound particle of mass \(m\) described by the wave function $$ \psi(x)=A x e^{-x^{2} / 2 b^{2}} $$ For a total energy of 0 , the potential energy is given in Exercise 96. (a) Given these, to what region of the x-axis would a classical particle be restricted? Is the quantum-mechanical particle similarly restricted? (b) Write an expression for the probability that the (quantum-mechanical) particle would be found in the classically forbidden region, leaving it in the form of an integral. (The integral cannot be evaluated in closed form.)

Exercises \(90-92\) refer to a particle described by the wave function $$ \psi(x)=\sqrt{\frac{2}{\pi}} a^{3 / 2} \frac{1}{x^{2}+a^{2}} $$ Show that the normalization constant is correct.
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