Q BOHR V HEISENBERG V SCHRODINGER

 A

Ah! Comparing Heisenberg and Schrödinger is basically comparing two fundamental formulations of quantum mechanics. Let’s break it down carefully and clearly.

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1. The Core Difference

Quantum mechanics can be described in different “pictures” or formalisms. The two most famous are:

Aspect	Heisenberg Picture	Schrödinger Picture
Main idea	Operators evolve in time; states are fixed	States evolve in time; operators are fixed
Mathematical focus	( \hat{A}(t) = U^\dagger(t)\hat{A}(0)U(t) )	(
Time evolution	( \frac{d\hat{A}}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}] + \left(\frac{\partial \hat{A}}{\partial t}\right) )	( i \hbar \frac{\partial}{\partial t}
Intuition	Observables (like position, momentum) “move”	The wavefunction “moves”

In short:

• Heisenberg: the observables change, the state vector is static.

• Schrödinger: the state changes, the observables are static.

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2. Historical Context

• Heisenberg (1925): Developed matrix mechanics, using matrices to represent physical quantities. Focused on observable quantities (like energy levels of atoms).

• Schrödinger (1926): Developed wave mechanics, using the wavefunction (\psi(x,t)) to describe probabilities. Provided a more visual and intuitive picture.

Both formulations were proven equivalent mathematically — they describe the same physics, just from different viewpoints.

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3. Examples

1. Free particle in 1D

   • Schrödinger: Solve ( i \hbar \frac{\partial \psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} )

   • Heisenberg: Track how (\hat{x}(t)) and (\hat{p}(t)) operators change over time using commutators.

2. Harmonic oscillator

   • Schrödinger: Wavefunction spreads and oscillates.

   • Heisenberg: Position (\hat{x}(t)) and momentum (\hat{p}(t)) oscillate sinusoidally, operators are dynamic.

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4. Why Both Matter

• Heisenberg picture is more natural in quantum field theory and many-body physics, where operators represent fields.

• Schrödinger picture is often more intuitive in quantum chemistry and single-particle systems, where wavefunctions are visualized.

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✅ Key Takeaway

They are two sides of the same coin. Heisenberg focuses on changing operators, Schrödinger focuses on changing states, but predictions for measurements are identical.

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A

Ah, now you want to compare Heisenberg (or general quantum mechanics) vs Bohr. Let’s break this down carefully because Bohr’s model and modern quantum mechanics are very different in approach.

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1. Bohr Model (1913)

Niels Bohr proposed a model of the atom before full quantum mechanics existed. Key features:

1. Electrons move in fixed orbits around the nucleus (like planets around the Sun).

2. Quantized angular momentum:
   [
   L = n \hbar, \quad n = 1,2,3,...
   ]
   Only certain orbits are allowed.

3. Energy levels are quantized: Electrons can “jump” between orbits by absorbing or emitting photons:
   [
   \Delta E = h\nu
   ]

4. Explained hydrogen spectrum very well.

Limitations:

• Only works well for hydrogen or hydrogen-like atoms.

• Cannot explain multi-electron atoms, fine structure, or electron spin.

• Electrons in Bohr’s model are still classical particles in fixed orbits, just with quantized angular momentum.

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2. Quantum Mechanics (Heisenberg / Schrödinger)

Modern quantum mechanics (1925–1926 onwards):

1. Electrons are not in fixed orbits — they are described by wavefunctions ((\psi)) or operators ((\hat{x}, \hat{p})).

2. Probability interpretation: The square of the wavefunction gives the probability density of finding an electron in space.

3. All quantum phenomena are included:

   • Spin

   • Multi-electron atoms

   • Tunneling

   • Quantum entanglement

4. Bohr’s quantized orbits emerge naturally as energy eigenstates in the Schrödinger equation for hydrogen.

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3. Side-by-Side Comparison

Feature	Bohr Model	Quantum Mechanics
Electron position	Fixed circular orbits	Probabilistic cloud (wavefunction)
Angular momentum	Quantized: (L=n\hbar)	Naturally quantized from Schrödinger equation
Energy levels	Quantized	Quantized (same as Bohr for hydrogen), but also explains fine structure
Applicability	Hydrogen-like atoms	All atoms and molecules
Physics	Semi-classical	Fully quantum, consistent with Heisenberg uncertainty principle
Time evolution	Electron “jumps” between orbits	State evolves continuously (Schrödinger) or operators evolve (Heisenberg)

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4. Key Takeaways

• Bohr = stepping stone to quantum mechanics.

• Bohr introduced quantization of angular momentum and explained hydrogen spectra.

• Quantum mechanics (Schrödinger, Heisenberg) = full theory describing electrons as probability waves rather than classical particles.

• Bohr’s results are recovered as a special case in modern quantum mechanics for simple systems.

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If you want, I can make a visual comparison of Bohr orbits vs Schrödinger wavefunctions for a hydrogen atom — it really highlights the difference.

Do you want me to do that?