← Quantum Mechanics
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Topic 1 of 12

From Classical to Quantum Theory

The historical experiments and crises that forced physics beyond the classical picture, and the first steps toward describing a quantum state.

The Infrared Crisis

The infrared crisis is arguably the first demonstration of quantum mechanics. Otto Lummer and Ernst Pringsheim performed experiments in 1892 that measured the blackbody spectrum. They showed that Wien's proposal did not accurately describe the spectrum at low frequencies (infrared).

Ultraviolet Catastrophe

The Ultraviolet catastrophe is known

Franck-Hertz Experiment

Content coming soon.

Stern-Gerlach Experiment

Content coming soon.

The State Ψ\Psi

There is a difference between the state ψ\psi and Ψ\Psi. ψ\psi is understood to mean the wave function in a specific basis, without regard to the universe of knowledge about the state. The state Ψ\Psi contains the universe of knowledge about the state. We have to tensor product different bases together in order to form the full knowledge about the state Ψ\Psi. For example, position and spin bases live in different Hilbert spaces (which is a fancy way of saying vector spaces), so if we have a state ψ\psi we can find out any information about ψ\psi that lives in the same vector space. However, if we want to discover information that lives in a different Hilbert space, such as momentum to spin, someone would have to tell us or we would have to know it ourselves. Now, we can combine our knowledge about these unrelated degrees of freedom using tensor products. For example, as we will learn in a little bit, we use the symbol H^\hat{H} to describe the total energy of the state Ψ\Psi. The state Ψ\Psi has energy both in the position and spin bases, so if we wanted to know the total energy, we would have to use the tensor product in order to connect these. H^=Hx^⊗H^spin\hat{H} = \hat{H_x} \otimes \hat{H}_{spin}

Types of Degrees of Freedom

There are spatial degrees of freedom, like position and momentum, and there are internal degrees of freedom, like spin.

Key Takeaways

  • The infrared and ultraviolet failures of classical blackbody theory were among the first cracks in classical physics
  • The state Ψ\Psi carries the full universe of knowledge about a system; ψ\psi describes it in one particular basis
  • Unrelated degrees of freedom (like position and spin) live in different Hilbert spaces and combine via tensor products
  • Degrees of freedom split into spatial (position, momentum) and internal (spin)