# Is the conjugate of a solution to the time-independent Schroedinger equation also a solution?

Was reading the second chapter of Griffith's Introduction to Quantum Mechanics and have failed to understand why the conjugate of a solution to the Time-Indepedent Schrodinger Equation (hence TISE) is a solution itself.

It's at the end of Section 2.1 for reference (contained in Problem 2.1). I have the second edition.

## 1 Answer

Consider the time-independent Schrodinger Equation (TISE) in position space:

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi.$$

Suppose we take the complex conjugate of both sides. This then gives us

$$-\frac{\hbar^2}{2m}\frac{d^2\psi^*}{dx^2} + V(x)\psi^* = E\psi^*,$$

since the potential and energy are clearly real, and the derivative is unaffected by conjugation. Therefore it is plain to see that if $\psi$ is a solution to the TISE, then so will its complex conjugate $\psi^*$.

• Thanks heaps @Jared Dziurgot ! That is so simple! I think I was reading into it too much. – Alister Chew Dec 19 '17 at 23:29
• Actually the potential could be imaginary if absorbtion or decay of particles were possible. – ZeroTheHero Dec 20 '17 at 0:48
{"ft_lang_label":"__label__en","ft_lang_prob":0.8945842,"math_prob":0.98362386,"size":319,"snap":"2019-35-2019-39","text_gpt3_token_len":75,"char_repetition_ratio":0.12063492,"word_repetition_ratio":0.0,"special_character_ratio":0.20689656,"punctuation_ratio":0.084745765,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997148,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-24T12:10:15Z\",\"WARC-Record-ID\":\"<urn:uuid:56511da1-6e66-4ed0-ba69-1d9c53bbdb69>\",\"Content-Length\":\"138706\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1fd81355-9962-472e-bc90-b30c89a504aa>\",\"WARC-Concurrent-To\":\"<urn:uuid:a20f1fab-476b-4688-be4f-b7eb9934ca74>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/375379/is-the-conjugate-of-a-solution-to-the-time-independent-schroedinger-equation-als?noredirect=1\",\"WARC-Payload-Digest\":\"sha1:MV5ANH5IBQSCNRFSRS3RIYXRTQOUVRJ6\",\"WARC-Block-Digest\":\"sha1:X237CUDP4LZBIWGCHLCQI2IX5CDTD6DA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027320734.85_warc_CC-MAIN-20190824105853-20190824131853-00161.warc.gz\"}"}