Understanding the Non-Contradiction in Quantum Mechanics: Total Energy of Superposition States

The time-independent Schr?dinger equation is a cornerstone in the field of quantum mechanics, providing a framework to understand the behavior of particles in various systems. However, it often raises questions when examining the energy states of quantum systems, particularly when dealing with superposition states. Let's delve into the principles that demystify the apparent contradiction where the total energy of a solution to the time-independent Schr?dinger equation can differ from those of the eigenstates.

The Principles of Quantum Mechanics

The time-independent Schr?dinger equation describes the stationary states of a quantum system, where solutions to the equation (eigenstates) correspond to specific energy eigenvalues. These eigenstates are definitive in terms of energy, reflecting the energy level of a quantum system.

Eigenstates and Their Energy

Each eigenstate of the Schr?dinger equation has a definite energy, which is determined by the eigenvalue associated with it. This definiteness is a fundamental characteristic of the eigenstate and underpins the principle of energy quantization in quantum systems. The eigenvalue represents a specific energy level that the quantum system can occupy when in this state.

Superposition States and Energy Superposition

A general quantum state can be a superposition of multiple eigenstates, meaning it is a linear combination of these eigenstates. This superposition state does not have a definite energy but rather a probability distribution over the energies of the individual eigenstates. The energy of such a superposition state cannot be precisely defined as a single value but instead is described probabilistically. This is because the state is a combination of different energy levels, each with its own probability amplitude.

Energy Conservation and Superposition States

When discussing the principle of energy conservation in quantum mechanics, it is important to clarify that energy is indeed conserved, but it is not a dynamical variable in the same way as position or momentum. The conservation principle implies that the total energy of a closed system remains constant over time. However, in the case of superposition states, the energy is not a single, definite value but rather a distribution over the possible eigenvalues. This distribution reflects the quantum nature of the system and is consistent with the quantum mechanical principles.

Practical Implications and Spectra

The time-independent Schr?dinger equation allows for a range of solutions with different energies, corresponding to different eigenvalues. These solutions describe the possible energy states of the system. An electron transitioning between these states via absorption or emission of energy is the basis for spectral lines. The electron cannot occupy any arbitrary state; it must transition between definite energy levels according to the laws of quantum mechanics. This principle of non-continuous energy levels is reflected in the discrete nature of the spectra observed in various quantum systems.

Conclusion

The total energy of a superposition state may not correspond to a single eigenvalue, which might initially seem contradictory. However, this non-contradiction is a fundamental aspect of quantum mechanics, where the energy of a superposition state is described by a probability distribution over the individual eigenvalues. The conservation of energy remains a core principle, albeit in a probabilistic form for superposition states. By understanding these principles, we can better grasp the complexities and beauty of quantum mechanics.

In summary, while the total energy of a superposition state may not correspond to a single eigenstate's energy, this aligns with the principles of quantum mechanics and does not constitute a contradiction. The non-contradictory nature of quantum mechanics is further supported by the conservation of energy and the distinct energy levels described by the eigenstates of the time-independent Schr?dinger equation.