Understanding The Math Behind Planck’s Law, The Equation That Started Quantum Mechanics.

Planck’s Law

In the first part of the fifth video of ‘Introduction to Quantum Mechanics’, we will discuss the derivation of Planck’s law. In the previous video, we learned what is a black body and how Rayleigh and Jeans, and Wein failed to explain the black body spectrum. They considered blackbody radiation to be made of standing waves inside a cubical box. Using classical statistical mechanics, they calculated the number of modes of vibrations or the number of standing waves inside a frequency interval. They associated average energy to each of the standing waves – kT where k is the Boltzmann’s constant and T is the temperature.

By multiplying this average energy with the number of modes, we get the energy density – the Rayleigh-Jeans law. However, the Rayleigh-Jeans law can only explain the black body spectrum at low frequencies. It blows up at high frequencies. This is known as the Ultraviolet Catastrophe.

The black body radiation spectrum was finally explained by German Physicist Max Planck. He assumed that the standing waves can only have a particular energy, in the multiples of hv where h is the Planck’s constant and v is the frequency. Using this assumption, Max Planck mathematically derived Planck’s radiation law that could easily explain the shape of the black body spectrum. We can also derive the Rayleigh-Jeans Law, Wein’s law, and Stefan’s law from Planck’s radiation law. This will be covered in the second part of the video.

Previous In Series: What Is Black Body Radiation And How Classical Physics Failed To Explain It?

3 thoughts on “Understanding The Math Behind Planck’s Law, The Equation That Started Quantum Mechanics.”

1. John Z Dill shown to be the actual driver of

Can you articulate this “shape” of the black body spectrum in plain English?

2. Israel Socratus Sadovnik

In 1900 Planck united together two formulas ( Rayleigh–Jeansfor for long and Wien’s for short wavelengths) and then divided them. He was himself very surprised when the result was found correct. . . . Maybe some thousands of physicists were satisfied with this result as the end of searching. But the great Max Planck asked himself: ”Why is this formula correct? ”, ”What does the result mean?”. And in his Nobel Lecture given on 2 June 1920, Planck described how he made his discoveries: ” . . . eventually after some weeks of the hardest work of my life, light entered the darkness, and a new inconceivable perspective opened up before me. … ” The result was – quantum of action (as energy multiplied time: h=Et). The coefficient (h) was neither in the Rayleigh–Jeansfor nor in the Wien’s formulas. Planck took unit (h) as in some books are written: “intuitively, instinctively, phenomenological” . . . . Planck didn’t explain MATH details of  (h) . . . He took (h)  ” from heaven ” . . . At first the theory met resistance, but . . . later  the existence of (h) was proved by other physicists.

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