The Schrödinger equation is just another way of writing the conservation of energy, right? So how can you use it to find the quantum wavefunction? I mean in every example I've seen the wavefunction is calculated using sines and cosines so I don't see how you can use the Schrödinger equation to find a wavefunction. All I can see it being useful in is finding energy levels which doesn't help calculate the wavefunction. I'm pretty sure I'm wrong about everything I said because the Schrödinger equation is a big deal in QM, so can you please explain me what this is actually used for? I've had this question for ages now. Any help would be helpful.
Quantum Mechanics – What is the Schrödinger Equation Used for Exactly?
quantum mechanicsschroedinger equationtime evolutionwavefunction
Related Solutions
Yes, your model can be fully solved analytically, though actually finding the spectrum may involve the numerical solution of a trascendental equation. This is done explicitly in several introductory QM textbooks, but in essence all you need to do is solve locally for each region (which gives you sines and cosines, or exponentials) and then stitch the solutions by demanding continuity of the wavefunction and its derivative.
In essence, as $a$ becomes bigger than the ground state energy, you end up solving for two weakly coupled wells, with roughly independent ground states; the global well ground and first excited state then become even and odd linear combinations of the two individual ground states, with an energy splitting that depends on the coupling between the two wells.
Regarding your other points,
- An other would be at what point this middle section becomes an infinite wall?
When the middle bit goes up to infinity.
- Where will the Ψ be equal to zero in x=0.5 in this particular potential function?
When the middle bit goes up to infinity. The ground state is never zero there (though it does decay exponentially in $a$); the first excited state is identically zero in the middle.
Well in QM conservation of energy is violated right?(nuclear explosions, nuclear energy, particle accelerators, the sun, other stars)
None of these things violate the conservation of energy. They merely change energy from one form (e.g. mass) to another form (e.g. electromagnetic). In all these cases the energy is the same before and after the interaction.
Then why does the Schrödinger equation depend on the conservation of energy?
The specific form that you posted is the time independent form. That is based on the Hamiltonian, which is more or less as you say, an expression of the total energy. According to Noether’s theorem the conservation of energy is directly tied to time translation invariance. Therefore, it should not be surprising that the time independent form features energy prominently.
Best Answer
Wrong. The original motivation for Erwin Schrödinger can be read in his 1926 paper An Undulatory Theory of the Mechanics of Atoms and Molecules (behind paywall, but freely readable copies can easily be found on the web).
Namely, the ansatz is the Hamilton-Jacobi equation, whose solution $W(x,y,z,t)$ is seen as a set of constant-value surfaces in $(x,y,z)$ propagating in space as time $t$ changes, one (possibly disjoint) surface for a chosen constant value. Reinterpreting these surfaces as the constant-phase surfaces, or wavefronts, of some waves, gives us the idea of the quantum wavefunction (whose actual interpretation was not known until Max Born formulated the rule named after him).
Now, when we try to solve the time-dependent Schrödinger's equation, we find that it can be in general variable-separated for the time dimension $t$ to yield an expansion of the wavefunction in eigenfunctions of the Hamiltonian operator. This then gives us the time-independent Schrödinger's equation for the individual eigenfunctions.
This is not much different from what we do when we solve e.g. the wave equation for oscillation of a drum-like membrane. The nodes of the eigenmodes of vibration of such a membrane form the famous Chladni figures.
The cases when you can form a wavefunction with merely sines and cosines are very rare. They are the most trivial types of problems that you can solve with Schrödinger's equation. Even the harmonic oscillator in quantum mechanical description requires the use of a set of functions that are definitely not sines nor cosines.
To understand how to use the Schrödinger's equation to find a wavefunction (I'm assuming you mean the eigenfunction of Hamiltonian, rather than time evolution of an arbitrary wavefunction), try reading the derivations of the solutions of various problems like quantum harmonic oscillator and hydrogen atom.