As I understand it, where a system’s Hamiltonian is not time-dependent, the wave function of a system that is in state psi(0) at time t=0 evolves as:(adsbygoogle = window.adsbygoogle || []).push({});

psi(t) = sum, over all eigenvalues E of operator H, of exp(-i*E*t / hbar) * <E|psi(0)> * | E>

If the eigenvalues are continuous it is an integral rather than a sum but let’s assume they are discrete.

In Shankar’s ‘Principles of Quantum Mechanics’, he claims on p118 (2nd edition) that multiplying a ket by a complex number of modulus one does not change the physical state. The first factor in the above formula has modulus one so, we can re-write the equation as:

psi(t) = sum, over all eigenvalues E of operator H, of <E|psi(0)> * f(t,|E>)

where f(t,|E>)=exp(-i*E*t / hbar) * | E> is a ket representing a state that is physically identical to |E>.

Since f(t,|E>) is physically identical to |E> and the first postulate of QM tells us that the ket represents the physical state, it should make no physical difference if we replace f(t,|E>) by |E> in the formula.

That then gives us:

psi(t) = sum, over all eigenvalues E of operator H, of <E|psi(0)> * |E>

But now we have a formula from which t has disappeared, which implies that the state does not change over time.

Where did I go wrong?

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# Question about evolution of a wave function

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