# 4: Quantum Mechanics

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- 47391

Quantum mechanics is a powerful framework for understanding the motions and interactions of particles at small scales, such as atoms and molecules. The ideas behind quantum mechanics often appear quite strange. In many ways, our everyday experience with the macroscopic physical world does not prepare us for the microscopic world of quantum mechanics. The purpose of this chapter is to introduce you to this exciting world.

- 4.1: Prelude to Quantum Mechanics
- The quantum-computer processor is the “brain” of a quantum computer that operates at near-absolute zero temperatures. Unlike a digital computer, which encodes information in binary digits (definite states of either zero or one), a quantum computer encodes information in quantum bits or qubits (mixed states of zero and one). Quantum computers are discussed in the first section of this chapter.

- 4.2: Wave functions
- In quantum mechanics, the state of a physical system is represented by a wave function. In Born’s interpretation, the square of the particle’s wave function represents the probability density of finding the particle around a specific location in space. Wave functions must first be normalized before using them to make predictions. The expectation value is the average value of a quantity that requires a wave function and an integration.

- 4.3: The Heisenberg Uncertainty Principle
- The Heisenberg uncertainty principle states that it is impossible to simultaneously measure the x-components of position and of momentum of a particle with an arbitrarily high precision. The product of experimental uncertainties is always larger than or equal to \(\frac{\hbar}{2}\). The energy-time uncertainty principle expresses the experimental observation that a quantum state that exists only for a short time cannot have a definite energy.

- 4.4: The Schrӧdinger Equation
- The Schrӧdinger equation is the fundamental equation of wave quantum mechanics. It allows us to make predictions about wave functions. When a particle moves in a time-independent potential, a solution of the time-dependent Schrӧdinger equation is a product of a time-independent wave function and a time-modulation factor. The Schrӧdinger equation can be applied to many physical situations.

- 4.5: The Quantum Particle in a Box
- 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.

- 4.6: The Quantum Harmonic Oscillator
- The quantum harmonic oscillator is a model built in analogy with the model of a classical harmonic oscillator. It models the behavior of many physical systems, such as molecular vibrations or wave packets in quantum optics. The allowed energies of a quantum oscillator are discrete and evenly spaced. The energy spacing is equal to Planck’s energy quantum. The ground state energy is larger than zero. This means that, unlike a classical oscillator, a quantum oscillator is never at rest.

- 4.7: Quantum Tunneling of Particles through Potential Barriers
- A quantum particle that is incident on a potential barrier of a finite width and height may cross the barrier and appear on its other side. This phenomenon is called ‘quantum tunneling.’ It does not have a classical analog. The tunneling probability is a ratio of squared amplitudes of the wave past the barrier to the incident wave.

Thumbnail: Schrödinger took the absurd implications of this thought experiment (a cat simultaneously dead and alive) as an argument against the Copenhagen interpretation. However, this interpretation remains the most commonly taught view of quantum mechanics.

## Contributors and Attributions

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).