# Stupid question about superposition of quantum states

• Robsta
In summary, a quantum-mechanical harmonic oscillator with frequency ω has Hamiltonian eigenstates and eigenvalues. The state of the oscillator evolves as a function of time t, and the first time for which the time-evolved state is orthogonal to the initial one can be calculated using the inner product in bra-ket notation. The probability of finding the system in state |0> at time t is 1/2, and the time dependence disappears when taking the modulus square. However, for the first time when the state is orthogonal to the initial one, the inner product must equal 0.
Robsta

## Homework Statement

A quantum-mechanical harmonic oscillator with frequency ω has Hamiltonian eigenstates |n with eigenvalues En = (n + 1/2) ħω. Initially, the oscillator is in the state (|0> + |1>)/√2. Write down how the state of the oscillator evolves as a function of time t. Calculate the first time for which the time-evolved state is orthogonal to the initial one.

## The Attempt at a Solution

I know how to evolve the states in time using the exponential solution to the TDSE.

|ψ(t)> = (e-iωt/2|0> + e-3iωt/2|1>)/√2.

This is fine. now I want the probability of finding the system in state |0> at time t. I bra through with <0|
<0|ψ(t)> = (e-iωt/2<0|0> + e-3iωt/2<0|1>)/√2.

<0|1> are orthogonal states so their inner product dissapears. <0|0> = 1

<0|ψ(t)> = e-iωt/2/√2

The probability of finding the state is |<0|ψ(t)>|2

P(|0>) = 1/2

Now here's the problem. When I take the mod square, the time dependence disappears, implying that the probability is constant for all time. This isn't true, at least I don't think it's true. Isn't the probability supposed to oscillate? I've forgotten how this oscillation comes about in bra-ket notation.

Robsta said:

## Homework Statement

A quantum-mechanical harmonic oscillator with frequency ω has Hamiltonian eigenstates |n with eigenvalues En = (n + 1/2) ħω. Initially, the oscillator is in the state (|0> + |1>)/√2. Write down how the state of the oscillator evolves as a function of time t. Calculate the first time for which the time-evolved state is orthogonal to the initial one.

## The Attempt at a Solution

I know how to evolve the states in time using the exponential solution to the TDSE.

|ψ(t)> = (e-iωt/2|0> + e-3iωt/2|1>)/√2.

This is fine. now I want the probability of finding the system in state |0> at time t. I bra through with <0|
<0|ψ(t)> = (e-iωt/2<0|0> + e-3iωt/2<0|1>)/√2.

<0|1> are orthogonal states so their inner product dissapears. <0|0> = 1

<0|ψ(t)> = e-iωt/2/√2

The probability of finding the state is |<0|ψ(t)>|2

P(|0>) = 1/2

Now here's the problem. When I take the mod square, the time dependence disappears, implying that the probability is constant for all time. This isn't true, at least I don't think it's true. Isn't the probability supposed to oscillate? I've forgotten how this oscillation comes about in bra-ket notation.
There is no problem with your calculation.

However, this is not what they are asking: they are asking the first time at which the state is orthogonal to the initial one so you need $\Bigl| \langle \psi(0) | \psi(t) \rangle \Bigr|^2$

## 1. What is superposition of quantum states?

Superposition of quantum states is a fundamental principle in quantum mechanics where a quantum system can exist in multiple states at the same time. This means that the system is in a state of being both "here" and "there" simultaneously.

## 2. How does superposition of quantum states work?

Superposition of quantum states works by combining two or more quantum states together to create a new state. This new state is a combination of all the original states, and the system will exist in all of these states simultaneously until it is observed or measured.

## 3. What is the significance of superposition of quantum states?

The significance of superposition of quantum states lies in its ability to explain and predict the behavior of particles at the subatomic level. It has also led to the development of technologies such as quantum computing and quantum cryptography.

## 4. Is superposition of quantum states proven?

Yes, superposition of quantum states is a well-established principle in quantum mechanics that has been proven through numerous experiments and observations. It is a fundamental aspect of our understanding of the behavior of particles at the quantum level.

## 5. Can superposition of quantum states be observed in everyday life?

No, superposition of quantum states is a phenomenon that only occurs at the subatomic level and cannot be observed in everyday life. It is a unique property of quantum particles and does not apply to larger, classical objects.

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